President Obama’s first meeting with Pope Francis produced a little schism of its own.

The Vatican and White House gave starkly different versions Thursday of Mr. Obama’s meeting with Francis.

The president’s account downplayed the Catholic Church’s concerns about religious freedom in the United States and Obamacare’s mandate to pay for contraception.

The pontiff and the president were cordial in the televised portions of their meeting, but a subtle competition to set the agenda played out after the meeting, which went well beyond its scheduled half-hour.

“We actually didn’t talk a whole lot about social schisms in my conversations with His Holiness,” Mr. Obama said at a press conference in Rome. “In fact, that really was not a topic of conversation.”

Mr. Obama deflected a reporter’s question about the extent of his discussion with the pope on the contraceptive mandate by saying that Francis “actually did not touch in detail” on the subject. The administration has been locked in a lengthy legal and political battle with the U.S. Catholic Church hierarchy over Obamacare and issues such as abortion and same-sex marriage.

The Vatican, however, issued a statement after the meeting saying the president’s discussions with Francis and two other top Vatican officials focused “on questions of particular relevance for the [Catholic] Church in [the United States], such as the exercise of the rights to religious freedom, life and conscientious objection” — issues that have fueled divisions between Mr. Obama and the church.

Although Mr. Obama wanted to highlight his bond with Francis over questions of economic inequality and helping the poor, Obamacare’s mandate for employers to pay for birth control gained more attention.

The president clearly wanted to benefit from the global popularity of the pope. Their meeting was a highlight of Mr. Obama’s foreign trip that ends Friday in Saudi Arabia, but it was at an awkward time for the president.

On Tuesday, the Supreme Court heard oral arguments on the Affordable Care Act’s mandate requiring for-profit employers of a certain size to offer insurance benefits that cover birth control and other reproductive health services without a co-pay. Some employers object to the mandate on the grounds that it violates their religious beliefs.

The president acknowledged that his meeting with Cardinal Pietro Parolin, the Vatican’s secretary of state, included a discussion about “the issue of making sure that conscience and religious freedom was observed in the context of applying the law.” Mr. Obama repeated the administration’s argument that it has tried to accommodate churches and faith-affiliated institutions in regard to the health care law.

“I explained to him that most religious organizations are entirely exempt,” Mr. Obama said. “Religiously affiliated hospitals or universities or [nongovernmental organizations] simply have to attest that they have a religious objection, in which case they are not required to provide contraception — although employees of theirs who choose are able to obtain it through the insurance company.”

Mr. Obama said he pledged to the cardinal “to continue to dialogue with the U.S. Conference of [Catholic] Bishops to make sure that we can strike the right balance — making sure that not only everybody has health care but families and women in particular are able to enjoy the kind of health care coverage that the ACA offers, but that religious freedom is still observed.”

The president said he found an ally in Francis, a native of Argentina, on the issue of immigration reform during their nearly hourlong discussion.

“As someone who came from Latin America, I think he was very mindful of the plight of so many immigrants who are wonderful people, working hard, making contributions, [and] many of their children are U.S. citizens,” Mr. Obama said. “And yet they still live in the shadows, in many cases have been deported and are separated from families.”

The president said he told Francis “how I felt that there was still an opportunity to make this right and get a law passed.”

House Republicans have resisted a Senate-passed comprehensive measure that would provide a pathway to citizenship for most of the 11 million illegal immigrants in the U.S., and instead have looked for a more piecemeal approach to reform.

Francis presented the president with two medallions, including one that symbolized the need for solidarity and peace between the Northern and Southern hemispheres, and a copy of his teaching titled “Evangelii Gaudium,” or “The Joy of the Gospel.”

Mr. Obama told the pope, “I actually will probably read this in the Oval Office when I’m deeply frustrated. I’m sure it will give me strength and calm me down.”

“I hope,” Francis replied.

The president gave Francis a custom-made chest with a variety of fruit and vegetable seeds used in the White House Garden. The pontiff announced this month that he would open to the public the gardens of the papal summer residence, the Apostolic Palace of Castel Gandolfo.

Mr. Obama asked Francis to bless his family. “They’ve been very strong. Pray for them. I would appreciate it,” Mr. Obama said. Francis assented.

White House National Security Adviser Susan E. Rice, who also attended the meeting, later told a reporter that the event was “really cool.”

The White House said that, in the spirit of the president’s gift, Thomas Jefferson’s Monticello will donate seeds that will yield several tons of produce to a charity of Francis’ choosing.

“These, I think, are carrots,” Mr. Obama told Francis as he displayed a seed pouch. “Each one has a different seed in it. The box is made from timber from the first cathedral to open in the United States in Baltimore.”

Mr. Obama also told Francis, “If you have a chance to come to the White House, we can show you our garden as well.”

Francis responded in Spanish, “Como no,” which translates to “Why not?” or “For sure.”

The president often expresses admiration for Francis, who recently celebrated his first year in the papacy. Francis at times has clashed openly with Mr. Obama, as when he led a vigil last year in opposition to the president’s plan to launch military strikes against Syria. But the pope has proved popular with liberal Catholics, who say he has emphasized the church’s social mission to aid the poor over divisive issues such as abortion.

The White House said the seed chest is made from American leather and features reclaimed wood from the Basilica of the National Shrine of the Assumption of the Blessed Virgin Mary.

The basilica’s cornerstone was laid in Baltimore by John Carroll, a Jesuit and the first Catholic bishop and archbishop in the United States.

The president included the stop in Italy after meetings with European leaders in Brussels on the crisis in Ukraine and other matters.

Before nightfall in Rome, Mr. Obama got a private tour of the Colosseum, the ancient Roman amphitheater where gladiators battled and Christians were executed. The president marveled that the structure is “bigger than some current baseball stadiums.”

As he walked with his tour guide, Mr. Obama asked her, “Let’s say someone is on the ground. He didn’t have to kill them, did he?”

Reporters couldn’t hear the guide’s answer.

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