WASHINGTON — President Biden was blunt. Democrats had to rally behind his $1.85 trillion economic and environmental spending bill, he told them on Thursday, because nothing less than his presidency was at stake.

“I don’t think it’s hyperbole,” he said as he unveiled a revised proposal and pleaded with Democratic lawmakers to support it during a last-minute morning meeting at the Capitol, hours before he left for a six-day trip to Europe to meet with world leaders.

“The House and Senate majorities and my presidency will be determined by what happens in the next week,” Mr. Biden told the lawmakers during the hourlong session, according to a person who was at the meeting.

The president’s proposals, while about half as costly as his original plan, still amount to a transformative agenda that would touch the lives of millions of Americans and serve as the core of his party’s argument to stay in power through the 2022 midterm elections and the 2024 presidential contest.

And even as party members have engaged in a fierce, ideological debate among themselves, the monthslong negotiation has thrown into stark relief the differences between Democrats and Republicans, almost all of whom have refused to back spending on child care, climate change, preschool, expanded Medicare services, free community college or higher taxes on corporations and the wealthy.

Mr. Biden and his aides gambled on Thursday, effectively calling for a final decision on his economic and environmental agenda and daring holdout Democrats not to back it. Senior administration officials said that the decision to go all-in was a product of the president’s belief that he had exhausted all avenues in the talks and secured the best possible package he could — and, crucially, that the package could command support from all corners of a fickle Democratic caucus.

But as he prepared to land in Rome, Mr. Biden’s bet had not yet paid off. He had not ended months of intraparty squabbling that has dragged down his poll ratings, jeopardized Democratic candidates and raised deep doubts among Americans that his presidency can deliver on the promises of a vast social and economic agenda.

In the closed-door session on Thursday, Speaker Nancy Pelosi told Democratic lawmakers that “when the president gets off that plane, we want him to have a vote of confidence from this Congress.” She urged them to vote on Thursday on a separate, bipartisan $1 trillion infrastructure measure that progressives have seen as their best leverage to ensure passage of the rest of Mr. Biden’s agenda.

Instead, for the second time in a month, Ms. Pelosi pulled back from plans on that vote after progressive Democrats objected again. They ignored the president’s entreaties, signaling their continued mistrust of moderate Democratic senators, whom they fear will not back Mr. Biden’s larger social spending bill when it finally comes to a vote.

Senior White House officials shrugged off the setback, saying the president’s formal request on Thursday set in motion the final act of a monthslong political drama. They expressed confidence that votes on both bills would happen soon. The bickering among Democrats would fade, one senior official said, when Americans started seeing the benefits of Mr. Biden’s plans, like when the administration breaks ground next year on new electric vehicle charging stations. The official asked for anonymity to speak about closed-door negotiations.

Administration officials also said they were not surprised by the public comments from Senators Joe Manchin III of West Virginia and Kyrsten Sinema of Arizona, moderate Democrats who had forced the original $3.5 trillion proposal to be halved. The two delivered halfhearted statements that pointedly did not promise that they would support the president’s new framework for a deal on the spending bill.

But White House officials concluded that it was time for Mr. Biden to put down his final marker, explicitly asking Democratic lawmakers for their support on a specific proposal. Having the president leave for a week on his trip without doing so would have left the process in limbo, administration officials said.

And yet, the legislative disarray of the moment had the potential to leave Mr. Biden no better off than he had been 24 hours earlier. He was set to arrive in Rome without tangible evidence that he could break the political logjam that has stalled progress on his promises. He had only the outlines of an agreement, with no firm proof that it would pass. It will fall to him in several days of meetings this weekend to persuade world leaders that he will prevail with his plans for corporate taxation, climate change and more.

The president’s agenda might eventually make its way to his desk. Lawmakers said they planned to continue working throughout the weekend toward votes on both bills. But in the meantime, Mr. Biden is left without a concrete plan that has the support of Congress to present at the G20 gathering or the climate change summit next week.

Still, he appeared to reach a critical juncture on Thursday on the strategy for his agenda, which he has pursued for months. The president initially proposed trillions in spending to overhaul the government’s role in the economy, but he has consistently said he is willing to compromise.

That challenge has required a delicate balance in his own party, which controls Congress by razor-thin margins. Mr. Biden first had to negotiate with Republicans on an infrastructure bill, largely to unlock support from Senate centrists on a larger spending bill that was meant to carry the portions of his agenda that could not win bipartisan support. He then had to balance the concerns of centrists, who worried about spending and taxing too much in the larger bill, with the complaints of progressives who wanted him to spend trillions more than he was ultimately able to get.

Bringing the Democratic Party together took months. Mr. Biden pushed centrists to come up from their original demands that the bill cost $1.5 trillion or less. He also pushed progressives to compromise for far less than they had hoped, and to jettison programs that Mr. Manchin and Ms. Sinema opposed.

Officials suggested that shortly before leaving for Europe, Mr. Biden had reached a natural conclusion of those discussions: He had pushed the centrists to come up as far as they could, they said, and was making the case to progressives that there would not be a better possible deal.

Mr. Biden started Thursday by unveiling what White House officials said was a detailed outline for the spending bill, telling reporters that the administration was “confident that this historic framework will earn the support of every Democratic senator and pass the House.” Even as the president pitched the plan to lawmakers, his chief of staff was hailing it as “transformational,” and interest groups were congratulating Mr. Biden.

Back at the White House, with Marine One roaring on the South Lawn to whisk him off on his trip, Mr. Biden said he had secured “a historic economic framework” and suggested that the time for bickering over the details was over. “No one got everything they wanted, including me,” he said. “But that’s what compromise is. That’s consensus. And that’s what I ran on.”

But in a rebuke that played out over the next several hours, Democrats refused to immediately come together behind the leader of their party.

And Mr. Manchin and Ms. Sinema remained steadfast in their opposition to including parts of the progressive agenda — like free community college, a large Medicare expansion and tax rate hikes on the wealthy — in the president’s social policy legislation. They made clear that they would not be swayed by the refusal of the progressive members to vote for the infrastructure measure.

Even as the day started, Mr. Biden seemed to sense the tensions.

In his remarks at the White House after the meeting with lawmakers, the president did not deliver the do-or-die message that he did behind closed doors. Instead, he hailed his framework as the logical result of “compromise” and “consensus” and delivered a version of the detailed speech he has been giving for months.

“That decision alone to invest in our children and their families was a major part of why we were able to lead the world for much of the 20th century,” Mr. Biden said of landmark spending by previous administrations. “But somewhere along the way, we stopped investing in ourselves, investing in our people.”

His pitch to lawmakers was simple, and personal: He made promises to voters during the 2020 presidential campaign, and now it was time to make good on them.

“This agenda — the agenda that’s in these bills — is what 81 million Americans voted for,” the president said. “More people voted than at any time in American history. That’s what they voted for. Their voices deserve to be heard, not denied — or, worse, ignored.”

Michael D. Shear reported from Washington, and Jim Tankersley from Rome.