“We can use peace,” Mr. Trump said.
Francis replied, “It is with all hope that you may become an olive tree to make peace.”
As he bade the pope farewell, Mr. Trump told him, “I won’t forget what you said.”
There was a sense in the Vatican that Mr. Trump was easier to talk to than his tough language on the campaign trail and sharp words toward Francis had led them to believe. “Trump’s bark is worse than his bite,” said a senior Vatican official who was granted anonymity because he was not authorized to speak about the meeting.
For Mr. Trump, who came here after stops in Saudi Arabia and Israel, the visit to the Vatican capped a tour of the ancestral homes of three of the world’s great monotheistic religions. For Francis, who made his own landmark visit to Egypt last month, it was a chance to welcome a second American leader, after President Barack Obama paid his respects in 2014.
Unlike that meeting, few expected a meeting of the minds. Pope Francis and Mr. Trump have diametrically opposed views on issues as varied as immigration, climate change and arms sales. Although both appeared determined not to let politics spoil their encounter, their fraught personal history and divergent personal styles made for a loaded backdrop.
At 8:20 a.m., under an azure sky, the president’s motorcade rolled into the courtyard of the Apostolic Palace. Ostrich-feather-plumed Swiss Guards stood at attention as Mr. Trump and his wife, Melania, stepped out of an armored Chevrolet sport utility vehicle — he in a dark suit, white shirt and black-and-white-stripe tie; she in a black dress with a veil on her hair. A few minutes earlier, the pope arrived in a lone blue Ford Focus. He stepped out and walked into a side entrance.
Archbishop Georg Gänswein, the prefect of the papal household, greeted Mr. Trump and escorted him to an antechamber outside the pope’s office, where, after a few seconds, Francis came to greet him.
“Thank you so much,” Mr. Trump said. “It’s an honor.”
After posing for a picture — “protocol,” the pope murmured — Mr. Trump took a seat across a wooden desk from Francis. Vatican officials shooed reporters out of the room and the two men met for half an hour (the pope’s session with Mr. Obama lasted 20 minutes longer).
Speaking to reporters later, Mr. Trump described the session as “fantastic.” Later, on Twitter, he wrote, “I leave the Vatican more determined than ever to pursue PEACE in our world.”
The Vatican said in a statement that the two discussed the Middle East and “the protection of Christian communities,” as well as “the joint commitment in favor of life, and freedom of worship and conscience” and the Catholic Church’s aid work on behalf of immigrants.
Immigration has been a fault line between history’s first South American pope and a president who came to power promising to build a wall to keep Mexican migrants out of the United States. But people close to the pope said in the days leading up to the meeting that he would not reprimand Mr. Trump, but seek to impart his values and build a dialogue that could cause a reconsideration of policies by Mr. Trump, if not a conversion.
A bell signaling the end of the audience rang at 9 a.m., and Mrs. Trump joined her husband and the pope. Francis looked graver than the beaming Mr. Trump, but he lightened up when he shook Mrs. Trump’s hand, jokingly asking her in Italian, “Did you give him potizza to eat?” (potizza is a Slovenian dessert).
Mr. Trump then introduced his daughter, Ivanka, who also wore a black dress and veil; his son-in-law, Jared Kushner; and other members of the American delegation, including Secretary of State Rex W. Tillerson and the national security adviser, Lt. Gen. H. R. McMaster.
Notably missing was Mr. Trump’s chief strategist, Stephen K. Bannon, who left the trip before coming to Rome. Mr. Bannon, a Catholic, has criticized Pope Francis as a socialist, a global elitist and promoter of Muslim migration to Europe.
“That is why this visit is so important,” said Jim Nicholson, a former American ambassador to the Vatican. “There still is an educational element to it. It is an opportunity for President Trump to learn a lot more about this man, his life and his formation.”
In his gifts, the pope seemed eager to impart a lesson. He gave the president a copy of his most recent World Day of Peace message (“I signed it personally for you,” Francis said), as well as three of his writings: on the family, the joy of the Gospel, and, most tellingly for a recipient who has called climate change a hoax, “Laudato Si’: On Care for Our Common Home.” Written in 2015, it is the first papal encyclical focused solely on the environment.
“Well, I’ll be reading them,” said Mr. Trump, who gave the pope a set of books by the Rev. Dr. Martin Luther King Jr.
The pope and the president were both elected as outsiders promising to carry the far-off voices of the forgotten to the centers of global power. But that is more or less where the similarities end.
Mr. Trump is the scion of a real estate developer and a thrice-married lover of all things gilded. Pope Francis has made a calling card out of modesty. When, in 2013, he paid his own hotel bill after being elected pope, Mr. Trump wrote on Twitter: “I don’t like seeing the Pope standing at the checkout counter (front desk) of a hotel in order to pay his bill. It’s not Pope-like!”
The two have fundamentally different views about how to restore balance to a global economic system they consider broken, with Mr. Trump focusing on the engines of capitalism and Francis fighting to protect the disadvantaged from the dehumanizing forces of the modern world.
But Francis is also a shrewd political operator, and he had said that he was not seeking a confrontation. “In our talk, things will come out,” he told reporters after a recent trip to Fátima, Portugal. “I will say what I think, he will say what he thinks, but I never, ever, wanted to make a judgment without hearing the person.”
That is a far cry from his previous remarks about Mr. Trump. In February 2016, Francis said, “A person who only thinks about building walls, wherever they may be, and not building bridges, is not Christian.”
Mr. Trump, a candidate at the time, swiftly returned fire. “For a religious leader to question a person’s faith is disgraceful,” he said at a campaign rally in South Carolina.
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