ROME — President Biden capped a long weekend of diplomacy on Sunday with a swaggering proclamation of America’s renewed force on the world stage, claiming credit for what he cast as breakthroughs on climate change, tax avoidance and Iran’s nuclear ambitions at the end of a Group of 20 summit that was missing some of his biggest global adversaries.Buoyed by a three-day return to the interpersonal negotiations that have defined his political career and still overcome emotionally by an extended Friday audience with Pope Francis, Mr. Biden shook off questions about his sagging poll numbers at home and projected new optimism for his teetering domestic policy agenda.He acknowledged contradictions and stumbling blocks to his long-term ambitions on issues like reducing greenhouse gas emissions with a smile. And he claimed significant progress from a summit that produced one large victory for his administration — the endorsement of a global pact to set minimum corporate tax rates — along with a deal between the United States and Europe that will lift tariffs including those on European steel and aluminum.In other areas, like climate change and restoring a nuclear accord with Iran, the summit produced few concrete actions.But the president told reporters repeatedly that the weekend had shown the power of American engagement on the world stage, and that it had renewed relationships that frayed under his predecessor, Donald J. Trump.“They listened,” Mr. Biden said. “Everyone sought me out. They wanted to know what our views were. We helped lead what happened here. The United States of America is the most critical part of this entire agenda and we did it.”In the course of his Roman holiday, Mr. Biden sought to patch up relations with the French over a soured submarine deal, to bask in the blessing of the tax deal that his administration pushed over the line after years of talks, and to galvanize more ambitious climate commitments ahead of a global conference in Glasgow, Scotland, that he was traveling to next.The president left behind the chaos and disappointments of Washington, where recent surveys show that voter disapproval is mounting over his performance in office and that Democrats remain divided over a pair of bills that would spend a combined $3 trillion to advance his wide-ranging domestic agenda. Polling conducted by NBC News shows that seven in 10 Americans and almost half of Democrats believe America is going in the wrong direction.But after days of indulging in backslapping diplomacy at a time when bipartisan cooperation is in short supply at home, Mr. Biden emerged for his news conference on Sunday professing hope that both bills would pass the House in the next week and playing down the polls.“The polls are going to go up and down and up and down,” Mr. Biden said. “Look at every other president. The same thing has happened. But that’s not why I ran.”One reason Mr. Biden sought the presidency, after more than four decades as a senator and vice president, was for meetings like the Group of 20, where he is able to practice the flesh-pressing politics he has long enjoyed.World leaders have been slow to reconvene in person as the pandemic has stretched into its second year, but Mr. Biden attended a Group of 7 meeting in England in June that was a diplomatic icebreaker of sorts for wealthy countries. The summit in Rome brought a larger group of leaders together, though some of Mr. Biden’s largest rivals on the world stage, like China’s Xi Jinping and Russia’s Vladimir Putin, stayed home.Mr. Biden and other world leaders said the return to in-person talks changed the dynamic.Mario Draghi, the Italian prime minister whose country hosted the summit, said at a news conference that attendees were more willing than they had been in the past to address climate change, inequality, and other problems that would require collective action to fix.“Something changed,” Mr. Draghi said.Mr. Biden had hourlong meetings at the summit with leaders of varying influence.The prime minister of Singapore, Lee Hsien Loong, got 80 minutes. On Sunday, Mr. Biden also met with President Recep Tayyip Erdogan of Turkey on the sidelines, emerging with the shared promise to keep engaging on a range of disagreements, largely in view of Turkey’s influence in several critical regions, including Syria, Afghanistan, Libya and the Eastern Mediterranean.Mr. Biden said there were no substitutes for “looking at someone straight in the eye when you’re trying to get something done.”But in many areas, the summit produced more rhetoric than action.An agreement reached by the leaders on Sunday pledged to end the financing of coal power plants in countries outside their own and to “pursue efforts” to keep the average global temperature rise to 1.5 degrees Celsius by the end of this century.“We remain committed to the Paris Agreement goal to hold the global average temperature increase well below 2°C and to pursue efforts to limit it to 1.5°C above preindustrial levels,” the leaders said in a statement.The lack of further progress angered activists and presaged the difficulties Mr. Biden might face when he attends a high-stakes climate convention in Glasgow beginning on Monday.Mr. Biden conceded the irony in another push he made at the summit — for oil- and gas-producing countries to ramp up production to push down driving and heating costs — at a time when he is also urging the world to turn away from fossil fuels. But he said that the transition from oil and gas to lower-emission alternatives would not happen immediately, and that he was seeking to insulate consumers from price shocks in the meantime.The summit’s climate commitments drew quick criticism from environmental activists. Jennifer Morgan, the executive director of Greenpeace International, called the agreement among the leaders “weak,” and said it “lacked ambition and vision.” Jörn Kalinski, a senior adviser at Oxfam, said it was “muted, unambitious, and lacking in concrete plans.”Mr. Biden offered only incremental progress on the issue of unsnarling global supply chains, which was the subject of a side meeting of 14 countries that he hosted Sunday afternoon. Mr. Biden announced he was signing an executive order on the defense stockpile that will “allow us to react and respond more quickly to shortfalls” in supply chains.He also unveiled a deal to roll back tariffs on European steel and aluminum, an accord between the United States and the European Union that he said would benefit American consumers and “prove to the world that democracies are taking on hard problems and delivering sound solutions.”There were no resolutions over a protracted dispute about Turkey’s purchase of the Russian S-400 air defense system. Mr. Erdogan has refused to step back from the purchase, despite sanctions and expulsion from a U.S. defense program to develop the F-35 stealth fighter jet. And Mr. Biden did not agree to allow Mr. Erdogan to purchase F-16 fighter jets to update its fleet with money it had already spent for the F-35s.But as his news conference wound down, the engagement Mr. Biden lingered on longest was the one that kicked off his trip: his meeting with Pope Francis.Asked by a reporter about criticism from some conservative American Catholics that public officials like Mr. Biden, who are Catholic but support legal access to abortion, should be denied communion, Mr. Biden said the issue and his meeting with the pope were “personal.”The pope, Mr. Biden had said on Friday, called him a “good Catholic” and said he should continue to receive communion.On Sunday, Mr. Biden launched into a long reflection on his relationship with Francis, and his admiration for him. He recounted how the pope had counseled his family after the death of Mr. Biden’s eldest son, Beau, a tragedy he equated with losing “a real part of my soul.”Choking up at moments, Mr. Biden said the pope had become “someone who has provided great solace for my family when my son died.”The two men, Mr. Biden added, keep in touch.He walked off the stage, taking no further questions.Carlotta Gall, Jason Horowitz, and Somini Sengupta contributed reporting.