Trump, midway through a grueling nine-day, maiden worldwide journey, called upon the pontiff in a private, 30-minute meeting laden with religious symbolism and ancient protocol.
"Two of the issues that are the most pressing things that the world is facing, right now", said Professor Mark Markuly, dean of Seattle University's School of Theology and Ministry.
Francis's gift included a signed copy of his recent World Day of Peace message and three writings, one of which was Laudato Si': On Care for Our Common Home, all about climate change. The Times notes it's "the first papal encyclical focused exclusively on the environment". The two world leaders exchanged gifts during a meeting that was reportedly cordial, marking a contrast to their war of words in 2016.
The pope, by turns dour and smiling, welcomed a more effusive president to the seat of a religion that claims more than 70 million followers in the United States.
"We can use peace", said the president, acknowledging the symbolism.
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Trump retorted that it was "disgraceful" for the pope to doubt his faith.
"Well, I'll be reading them", Mr Trump said. "I hope you do", Trump said.
"We went to the Pope", Trump told Tusk and Juncker.
The two stuck mainly to protocol, avoiding a public reprise of the barbs they aimed at each other during Trump's presidential campaign or the pope's thinly veiled critiques of Trump as a symbol of a dangerously reinvigorated nationalism. "It was an honor to be with the pope".
Outside of a grinning trump, first daughter Ivanka, first lady Melania and the pope himself are all stone faced and appear to be unimpressed.
Later that day, Trump spoke of his encounter with Francis in glowing terms during a meeting with Prime Minister Paolo Gentiloni.
"We had a fantastic meeting", the president said afterward.
Trump included close family members and some long-time aides in the meeting, in a gesture being interpreted as a demonstration of his commitment to loyalists. Also in the audience were Rex Tillerson and H.R. McMaster.
Pope and the US President have been in agreement on a need for Muslim leaders to do more against extremists in their own communities. But there are relatively few other areas where their views align.