WASHINGTON — President Biden on Thursday unveiled his outline for a $1.85 trillion social safety net and environmental bill, imploring Democrats to put aside their differences and embrace a plan to provide universal prekindergarten, generous support for child care costs and the largest investment ever to combat climate change.

But his appeal for Democrats to unite and hand him a long-delayed victory on his domestic agenda fell flat, as liberals demanded assurances that the package would survive before they would agree to an immediate vote on a separate $1 trillion infrastructure bill. That left Mr. Biden empty-handed as he departed for Europe, where he had hoped to point to progress on both measures as proof that American democracy still works.

By Thursday evening, with Mr. Biden heading for Rome aboard Air Force One, the House Progressive Caucus had slammed the door shut on prospects of a quick win.

“Members of our caucus will not vote for the infrastructure bill without the Build Back Better Act,” the group said in a joint statement, using the name of the president’s social policy and climate bill.

It would provide preschool for more than six million 3- and 4-year-olds, child care and health care subsidies, monthly payments for families with children and $555 billion for programs to wean Americans from fossil fuels.

The day’s drama at the Capitol at once fleshed out the details of legislation that Speaker Nancy Pelosi called “spectacular” and underscored the deep rifts and wells of distrust within the party that Mr. Biden and top Democrats have been unable to overcome.

The president joined House Democrats in the Capitol on Thursday morning, hoping that by personally detailing his framework and projecting confidence about a compromise, he could paper over outstanding disputes about what should be in the plan. Doing so, he hoped, would unstick the infrastructure bill, which passed the Senate in August but has been held captive in the House by liberals demanding their legislative priorities: social policy and climate change.

Mr. Biden delayed his departure for Europe to plead for a victory.

“We have a framework that will get 50 votes in the United States Senate,” Mr. Biden privately told House Democrats, according to a person familiar with his remarks. “I don’t think it’s hyperbole to say that the House and Senate majorities and my presidency will be determined by what happens in the next week.”

But while the president found plenty of support for his plan — whose major components were still being debated even as he presented it — he departed for Rome with both prongs of his domestic agenda still hanging in limbo in a divided Congress.

Liberals said they trusted Mr. Biden to deliver on his outline, but that sentiment did not extend to Senators Joe Manchin III of West Virginia and Kyrsten Sinema of Arizona, crucial centrist holdouts whose concerns have forced Democrats to whittle down the package and jettison some of the left’s top priorities. The pair privately told associates that they supported the president’s latest proposal, but neither publicly promised to back it. Representative Cori Bush, Democrat of Missouri, declared that she felt “a little bamboozled.”

Mr. Manchin and Ms. Sinema, in turn, privately maintained that they would not be bullied into accepting social policy and climate legislation by House liberals holding up their infrastructure bill. Ms. Pelosi flashed exasperation after relaying the highway and transit programs set to expire on Oct. 31 without an infrastructure vote. “This is professional,” she snapped. “Let’s do it in a timely fashion and not leave doubt.”

On Thursday night, the House instead approved a short-term extension of those programs by a vote of 358 to 59, a sign that passage of both the infrastructure bill and the domestic policy plan may be far-off.

The president’s framework, while considerably more modest than the cradle-to-grave expansion of the safety net that he initially envisioned, still could be legitimately hailed as transformational. Provisions for young children would offer a significant boost to middle-class families that have struggled for decades with economic uncertainty. Preschool would be made available to more than six million 3- and 4-year-olds. Subsidies for child care would limit costs to no more than 7 percent of income for most families. Funding for both of those provisions would last for six years.

It includes about $555 billion for programs to move Americans to electric vehicles and entice utilities away from natural gas and coal, representing what would be the largest federal investment in combating climate change. A new Civilian Climate Corps would be created.

And it would be paid for largely by the rich, in part through a new surtax on the income of multimillionaires and a 15 percent minimum tax on the largest corporations, levied on the profits they boast to their shareholders, not the earnings they downplay to the I.R.S.

In public remarks at the White House, Mr. Biden praised the plan as “historic.”

“No one got everything they wanted, including me,” he said in the East Room before departing for Rome. “But that’s what compromise is. That’s consensus. And that’s what I ran on.”

But it was not enough to break the progressive blockade of the infrastructure bill. Liberals accurately noted that some major provisions of the framework were still being negotiated, including tax measures to pay for it, prescription drug price controls and immigration measures.

House Democrats released 1,684 pages of what they said was the legislative text of the Build Back Better Act, but Senator Sheldon Whitehouse, Democrat of Rhode Island, told reporters that the majority leader, Chuck Schumer of New York, had given senators another week to fight for provisions left out of the framework.

And, liberals said, the two senators most likely to bring down the social policy measure, Ms. Sinema and Mr. Manchin, were less than solid in their commitments to vote yes.

“After months of productive, good-faith negotiations with President Biden and the White House, we have made significant progress on the proposed budget reconciliation package,” Ms. Sinema said in a statement.

Mr. Manchin said only, “It’s in the hands of the House.”

With his approval ratings slipping and the economy back in a pandemic-induced slowdown, Mr. Biden finds himself badly in need of a win. Nearly a month ago, he went to the Capitol facing the same impasse over the infrastructure bill, and called off a vote to give negotiations on the social policy and climate legislation more time.

He was back on Thursday in an even more precarious political situation. Inflation is rising, and growth slowed to 0.5 percent in the last quarter. A loss in the Virginia governor’s race on Tuesday would be seen as a harbinger for Democratic defeats in next year’s midterm elections.

“Do your job,” the Democratic candidate, Terry McAuliffe, told Congress on CNN on Thursday. “Get this passed.”

Mr. Biden told lawmakers final passage of the bipartisan bill to build roads, bridges and tunnels; promote rural broadband; and begin transitioning the country to electric vehicles would help convince European leaders that democracy can still work. “The rest of the world wonders whether we can function,” he privately told House Democrats.

“We badly need a vote on both of these measures,” Mr. Biden added.

But liberals wanted more than talking points on a plan that was clearly still unfinished even as Mr. Biden lauded its components.

“You have the outline of a very significant piece of legislation,” said Senator Bernie Sanders of Vermont, the Budget Committee chairman whose views hold sway with liberals in the House. “I want us to make it better.”

Democratic leaders had been keen to hand the president a victory to take abroad. The president planned to attend a climate summit on Monday in Scotland, where he hoped to point to the deal as evidence of the United States’ commitment to tackling climate change.

“When the president gets off that plane, we want him to have a vote of confidence from this Congress,” Ms. Pelosi told Democrats during the private meeting, according to the person familiar with the session.

Liberal members of the House and the Senate had plenty to celebrate in the president’s framework, even after their initial 10-year, $3.5 trillion blueprint was pared down to a package slightly more than half that size. Among the provisions that survived were federal programs for home health care and community care for older Americans and people with disabilities; $150 billion for rental assistance, home-buying help, public housing repairs and other affordable housing programs; and a generous extension of subsidized health insurance under the Affordable Care Act to the working poor in 12 states that have refused to expand Medicaid under the health law.

But there was also plenty to lament. The centerpiece of Mr. Biden’s climate policy — a measure to reward utilities for switching to renewable energy and to punish those that will not — was stripped out at Mr. Manchin’s insistence. One of the biggest social policies in the original package, a $500 billion federal paid family and medical leave benefit, is also gone.

An anticipated expansion of Medicare to cover vision, dental and hearing care was down to just hearing. An innovative tax on the wealth of billionaires was also out, in favor of a surtax on multimillionaires that would hit income but not their mountains of wealth.

The promise of two years of free community college would go unfulfilled, and the expanded child tax credit, passed in March to give most families a monthly check of up to $300 per child, would be extended only into 2023, not made permanent.

Representative Mark Pocan, Democrat of Wisconsin, singled out Mr. Manchin and Ms. Sinema as the reason compromises had to be made.

“It’s not everything that we had hoped for, not what the president had hoped for when he had a $6 trillion package,” he said. “But you know, the president has said repeatedly to us that he’s still committed to that agenda, so we’ll get this done now. As long as he’s president, as long as we try to build on to the Senate — maybe not have to worry just about two members all the time — we’ll have opportunities to get other parts of his agenda done as well.”

Catie Edmondson and Jim Tankersley contributed reporting.