Seven minutes go fast. So the University of Notre Dame student got right to the point during his meeting with the Rev. John I. Jenkins, the university’s president, one day last fall.

A senior majoring in philosophy and biology, the student shuffled in wearing pressed khakis and a button-down shirt. He fidgeted, sat on the edge of his chair, and delivered a confession: He no longer felt the presence of God.

His crisis of faith stemmed in part from studying biology, he told the president. He was struggling to relate to the religious conviction of his friends and had sought the counsel of Father Jenkins, a priest in the order of the Congregation of the Holy Cross, who has been Notre Dame’s president since 2005.

Father Jenkins, who was seeing his fourth student of the day, later said the meetings, which he hosts twice per semester in his office in the iconic, golden-domed Main Building, are both energizing and exhausting. On a couple of occasions this day the president, who was out late the night before meeting with a local group, discreetly stifled yawns while peppering the earnest, bright students with probing, sometimes tough questions as well as thoughtful encouragement. After seven minutes, an assistant would knock on the door to signal that it was time for the next student.

This appointment ran long, however, and Father Jenkins ignored the knocks. A trim 56, with graying, wavy hair, he took off his wire-rim glasses and reclined in his chair. As he sometimes does when considering a question, he tilted his head back and briefly closed his eyes, then shared his own experience of questioning his faith when he was a senior majoring in philosophy on this same campus.

His belief in God had not wavered, Father Jenkins said, but as an undergraduate he had wrestled with how to interpret some Christian theology. Speaking in a soft voice, he said the student’s doubts “may be a kind of invitation to think about it more deeply.”

As the two men talked, the student smiled and seemed to relax. Seated near a picture of Father Jenkins with Pope Benedict XVI, they delved into questions of moral instincts, materialism, and science. Father Jenkins recommended that the student read The Screwtape Letters, an exploration of faith written by C.S. Lewis. He also mentioned the example of St. Thomas Aquinas, the 13th-century priest and patron saint of Roman Catholic education, who incorporated Greek philosophy and empiricism in his teachings. Father Jenkins is an Aquinas expert and a professor of philosophy with two degrees from the University of Oxford.

As the student got up to leave, the president asked him to visit again, outside of office hours, to continue their discussion.

“Be patient with yourself as you struggle with this,” Notre Dame’s leader said, adding that sometimes people can “understand God, as far as we don’t understand God.”

Few college presidents serve as religious adviser and strategic planner for a major research university on the same afternoon. The dual responsibilities can be a burden for Father Jenkins, he and others here admit, but they are also an asset. As both priest and president, he sits atop a storied institution that is constantly wrestling with its Catholic character. And the lines Father Jenkins must toe are much the same as those Notre Dame grapples with in its evolving relationship with the church, higher education, and society.

A flashpoint was the commencement speech that President Obama gave here last year, one of the higher-profile challenges faced by a university president in recent years.

Some Notre Dame students and faculty members loudly criticized the invitation and, in particular, the university’s decision to grant an honorary law degree to President Obama, saying it was the wrong message for a Catholic university to send, given the president’s views on abortion policy and stem-cell research.

Richard W. Garnett, a professor and associate dean at Notre Dame’s law school, was a vocal opponent. “I don’t think the debate was about what is said at a Catholic university,” he argued. “It’s about what a Catholic university says.”

Students organized a group called ND Response in the months before Mr. Obama’s arrival and promised to boycott the commencement, planning a protest instead. The criticism often centered on Father Jenkins, and a Web campaign was organized to dissuade potential donors to the university. Some of the most pointed comments came from the bishop of the Diocese of Fort Wayne-South Bend, the Rev. John M. D’Arcy, who said Father Jenkins had failed to keep him in the loop on the decision to invite President Obama.

But virtually all the local dissent was respectful, particularly among students and professors. The protest’s leaders did not call for the removal of Father Jenkins. They said their complaints were not politically motivated but were about protecting Notre Dame’s commitment to Catholicism.

That nuance was hard to find as the controversy heated up in the national news media. Some students fretted that outside protesters toting graphic signs would detract from their commencement.

They were right to worry. The demonstrators showed up, as did a plane that circled overhead for days, towing a huge picture of an aborted fetus. Scores were arrested and charged with trespassing, including Randall A. Terry, an anti-abortion activist, and Norma L. McCorvey, the “Jane Roe” plaintiff in the Roe v. Wade lawsuit. Angry callers flooded phone lines to Father Jenkins’s office as well as those of the university’s public-affairs department. Some of the calls were threatening.

Grant K. Schmidt, president of Notre Dame’s student government, was interviewed by CNN on one of the campus’s many picturesque malls. Afterward, he says, he received hundreds of e-mail messages and calls, some from outsiders who wanted to enlist his help in getting students to protest the commencement. “My phone was just vibrating for a solid hour,” Mr. Schmidt says.

Also logging time on national television was Mary K. Daly, a senior, who led ND Response. She says that Father Jenkins could have handled the Obama invitation better but that she holds no resentment. In fact, she praised him for subsequently forming an anti-abortion committee at the university, which Father Jenkins said would “consider and recommend to me ways in which the university, informed by Catholic teaching, can support the sanctity of life.”

Ms. Daly says she will join Father Jenkins and other Notre Dame students in the national March for Life in Washington this month.

“We refuse to condemn the university or condemn Father Jenkins,” she says.

Away from the campus, though, many people condemned Father Jenkins. Some even made it personal.

An extreme example came from the pundit Ann Coulter, who, in an interview on Fox News, said, “More interesting than watching Obama give a speech for graduation, they should’ve had the administrators of Notre Dame on stage, taking a polygraph test on whether they believe in God.”

In an interview, Father Jenkins says some of the criticism stung. And while he understands and accepts that many people did not agree with him over the Obama invitation, sometimes those disagreements devolve into personal demonization, he says.

“If you stick your head above the parapet, then you’re going to be personally attacked. I just am not going to be cowed by that. I may be wrong. But I’m going to stand up for having serious dialogue.”

Many here say one of Notre Dame’s greatest strengths is its willingness to engage thorny, emotionally charged issues. What caught people here off-guard about the commencement furor was its politicization and the changed tone of news coverage, in which opinion and outrage now often reign.

Controversy itself is nothing new on this campus. For better or worse, people tend to have strong feelings about the nation’s most prominent religious-affiliated university. Notre Dame’s symbolic importance trumps any measure of prestige, with a rich history weighted by religious, ethnic, and class identity. It is also home to the most mythical of college sports teams—a once dominant football team whose coach the university recently fired, to the tune of millions of dollars and countless headlines.

Father Jenkins, paraphrasing a quip by Theodore M. Hesburgh, the legendary former president of Notre Dame, says about the football team: “Half the people love us. Half the people hate us. But they all watch us.”

Other Catholic colleges can only nip at Notre Dame’s heels. Georgetown University and Boston College are competitive in national academic rankings and research laurels, but the Fighting Irish top most lists. And Notre Dame has a deeper Catholic identity than any other major research university.

Notre Dame is also in an enviable financial position, thanks to conservative planning that Father Jenkins likens to that of a family business. During flush times, when the endowment hit $7-billion, the university reduced its annual payout to 3.5 percent. While Notre Dame took a hit in the recession, and its endowment dropped to $5.5-billion, the university increased its payout to ensure smooth budgeting. So as other colleges have retrenched, Notre Dame was able to hire 59 new faculty members for the fall semester, many of whom were highly sought.

Non-Catholics are increasing among Notre Dame’s faculty ranks, a controversial development in some circles. Yet several of last year’s hires say the university’s Catholic core is part of the lure.

“It’s freeing in a sense,” says Daniel M. Brinks, an associate professor of political science who came to Notre Dame last year from the University of Texas at Austin. Mr. Brinks, who is not Catholic, says the openness on campus about religious beliefs helps students grapple intellectually with questions of morality and ethics.

“It’s easy to have those conversations in the classrooms,” he says. “It’s just more expected here.”

In addition, Notre Dame and other religious-affiliated colleges have the advantage of being able to tap into a growing number of students who seek a spiritual component to their education. Nationally, about three-quarters of college students are searching for meaning or purpose in life, according to a 2006 study by the Higher Education Research Institute, at the University of California at Los Angeles. Notre Dame’s Catholic commitment to social justice is a further draw for many students.

But Father Jenkins and others here acknowledge that integrating the Catholic mission and a major research university’s priorities is rarely easy. Many days bring seemingly minor dilemmas (should business cards feature the cross?) that can snowball quickly into controversy or even national news. And Notre Dame must constantly weigh its traditional focus on undergraduates against its growing research ambition.

Sponsored research at Notre Dame, although still relatively small at $80-million per year, is up from $30-million a decade ago. And the university is seeding research with $80-million of its own money. But the shift in resources has raised complaints from some on the campus who worry about threats to the university’s traditional undergraduate focus.

Three words have long defined Notre Dame: Catholicism, football, and undergraduates. One challenge for the university’s leaders is to get alumni and its many followers in the broader Catholic world excited about research.

Another is preserving the university’s traditional role as a center of Catholic intellectual life while the church evolves and the nation’s Catholic center moves from Irish- and Italian-Americans in the Northeast toward Hispanics in the Southwest.

The priest-as-president must manage the university’s balancing act.

“It’s a huge burden precisely because the place matters,” says Mr. Garnett, of the law school.

Nathan O. Hatch, Wake Forest University’s president and Notre Dame’s former provost, says his former colleague in South Bend walks the line as well as possible. He calls Father Jenkins a man of great conviction, a serious philosopher, and “a stickler for academic quality.”

The third of 12 children born to Helen and Harry Jenkins Jr., John Jenkins grew up in Omaha, where he was the high-school prom king and swim-team captain. During a recent mass held at Notre Dame’s College of Science, he spoke about his father, a gastroenterologist and academic who was a faculty member at Creighton University’s medical school.

Standing before a couple hundred students and professors in a green vestment with yellow trim, the president said his father had a fundamental commitment to the scientific method and enjoyed debunking medical fads. But bedside manner was also important to his father, who died a few years ago. Combining science with moral and religious truths made his father a more effective doctor, Father Jenkins told them, and it can “make you more powerful scientists and people.”

Father Jenkins is so soft-spoken that he can be hard to hear in person. But friends and colleagues say it is a mistake to underestimate his tough streak.

“John can be very aggressive and very determined,” says John Affleck-Graves, the executive vice president, who has worked with Father Jenkins for decades at the university.

Notre Dame has a tradition of strong presidents, most notably Father Hesburgh, one of the 20th century’s most significant college leaders. He led Notre Dame for 35 years, and under his tenure the university had a direct line into the national consciousness of civil rights, the Vietnam War, academic freedom, and other wrenching debates.

Father Hesburgh’s presidency was hardly without controversy. While several U.S. presidents considered him a close confidant, he was too outspoken for President Richard M. Nixon, who in 1972 fired him as chairman of the U.S. Commission on Civil Rights.

Now 92, Father Hesburgh attended Notre Dame’s commencement last May. During the ceremony, the university gave President Obama a framed photograph, taken at a civil-rights rally in 1964, in which Father Hesburgh is seen linking hands with the Rev. Martin Luther King Jr.

Father Jenkins, however, had never met this U.S. president. And some people worried that he might be too reserved to deliver on such a big stage.

For his part, Father Jenkins admits that he felt pressure as that day approached. He says Father Hesburgh helped ease his mind and even wrote a letter to his mother before the speech, saying her son was doing a good job. Father Jenkins says he spent many hours crafting the words he would deliver before introducing President Obama.

“I did work extremely hard on that speech,” he says. “I knew it was perhaps the most important speech I would ever give.”

On that Sunday, network-television cameras were rolling as he walked to the podium, which was emblazoned with the seal of the President of the United States.

He began shakily, stumbling over the word “faculty” in his greeting. But his voice grew in strength and cadence as he addressed the quiet crowd.

“Even among those who did not go to Notre Dame, even among those who do not share the Catholic faith, there is a special hope, a special expectation for what Notre Dame can accomplish in the world,” said Father Jenkins.

As the 14-minute speech built toward its conclusion, he directly addressed the controversy. In doing so he took several confident strides toward assuming the stature of his eminent predecessor as a strong voice in American higher education.

“Most of the debate has centered on Notre Dame’s decision to invite and honor the president,” Father Jenkins said. “Less attention has been focused on the president’s decision to accept. President Obama has come to Notre Dame, though he knows full well that we are fully supportive of the church’s teaching on the sanctity of human life, and that we oppose his policies on abortion and embryonic-stem-cell research,” he said, provoking a loud round of applause and cheers.

“Others might have avoided this venue for that reason. But President Obama is not someone who stops talking to those who differ with him,” Notre Dame’s president said, his voice cracking slightly, before he craned his neck to look directly at President Obama. “Mr. President, this is a principle we share.”

Another roar followed.

Correction (2/27/2015, 1:34 p.m.): This article originally misstated the year that President Nixon ousted Father Hesburgh as chairman of the U.S. Commission on Civil Rights. It was 1972, not 1969. The article has been updated to reflect this correction, which was discovered in the course of preparing an obituary for Father Hesburgh.