House Speaker John A. Boehner threw efforts to avoid the year-end “fiscal cliff” into chaos late Thursday, as he abruptly shuttered the House for the holidays after failing to win support from his fellow Republicans for a plan to let tax rates rise for millionaires.

The proposal — Boehner’s alternative to negotiating a broader package with President Obama — would have protected the vast majority of Americans from significant tax increases set to take effect next year. But because it also would have permitted tax rates to rise for about 400,000 extremely wealthy families, conservatives balked, leaving Boehner (Ohio) humiliated and his negotiating power immeasurably weakened.

No one could say late Thursday what will happen next. Just 11 days remain until the new year, when more than $500 billion in automatic tax increases and spending cuts will begin to take effect, threatening to undermine the sluggish recovery and prompt a new recession.

As a grim-faced Boehner hurried from the Capitol on Thursday evening, his office issued a statement abdicating responsibility for solving the crisis to Obama and the Democratic-controlled Senate.

“The House did not take up the tax measure today because it did not have sufficient support from our members to pass. Now it is up to the president to work with Reid on legislation to avert the fiscal cliff,” the statement said, referring to Senate Majority Leader Harry M. Reid (D-Nev.).

What going over the 'fiscal cliff' would mean . . .

White House press secretary Jay Carney vowed that Obama would act but offered no clear path through the political mine field. The House is now out of session until after Christmas. The Senate is scheduled to meet for only a few hours on Friday afternoon before members leave town until next Thursday.

“The president’s main priority is to ensure that taxes don’t go up on 98 percent of Americans and 97 percent of small businesses in just a few short days,” Carney said in a written statement. “The President will work with Congress to get this done and we are hopeful that we will be able to find a bipartisan solution quickly that protects the middle class and our economy.”

Senior Democrats, meanwhile, urged Boehner to stop wooing his hard-liners and resume negotiations with the White House over a debt-reduction compromise that could win the support of Democrats and Republicans.

“Speaker Boehner’s partisan approach wasted an entire week and pushed middle-class families closer to the edge,” Reid spokesman Adam Jentleson said in a statement. “The only way to avoid the cliff altogether is for Speaker Boehner to return to negotiations, and work with President Obama and the Senate to forge a bipartisan deal.”

Even if the resumption of talks were to yield a breakthrough, Boehner’s ability to deliver GOP votes for any proposal that raises taxes is now in doubt after he was unable to rally House Republicans behind what he had dubbed “Plan B.”

Emboldened liberals quickly argued that Democrats should demand additional concessions from Republicans, either upping the demand for fresh tax revenue or withdrawing Obama’s offer to seek savings through cuts in federal health and retirement programs.

The vote on Plan B was perhaps the most consequential test of Boehner’s leadership since he took control of the House early last year. Persuading a majority of Republicans to cast a politically treacherous vote to allow higher taxes could have enhanced his leverage with Obama in future talks to rein in the national debt, Republicans said. But failure could imperil his hold on power, said Craig Shirley, a Republican consultant who wrote a biography of former president Ronald Reagan.

“If this was a parliamentary system, tonight’s dissent on Plan B would be seen as a vote of no confidence in Boehner,” Shirley said. “The national GOP is now simply a collection of warring tribal factions.”

Until earlier this week, Boehner had been pursuing an entirely different course, trying to strike a deal with Obama to save $2 trillion over the next decade. Boehner offered to raise $1 trillion in fresh revenue, in part by increasing tax rates for millionaires, and he offered to delay a fight over the federal debt limit for one year — both major GOP concessions.

But then on Monday, he abruptly switched gears, complaining that the president had not made concessions of equal size and significance to control the soaring cost of federal health and retirement programs. Obama also insisted on delaying a debt-limit fight for two years, a position Republican leaders considered unacceptable.

Plan B was always a high-risk strategy. House leaders sold it as a $3.9 trillion tax cut — one of the largest in U.S. history. But it would have allowed taxes to rise by about $300 billion over the next decade compared with current law. Some conservatives labeled it a betrayal that would put congressional Republicans on record in support of a tax increase for the first time in more than two decades — while doing nothing to curb government spending.

As the week wore on and House leaders scrambled for votes, Plan B appeared to gain traction as a preferable alternative to a deal with Obama.

“The Republicans are in an untenable situation,” said Rep. Dan Burton (Ind.), who was leaning toward supporting the measure Thursday. “If we don’t do anything, we go over the fiscal cliff. And then the president will come back . . . and if the economy goes to hell, he’s going to say it was the Republicans’ fault.”

But Rep. Allen West (R-Fla.), serving his final days after losing his reelection bid in November, said the GOP brand would suffer if Republicans took a step that they had long argued amounted to a tax increase. “If you don’t draw a contrast with the other side, who are you?” he asked.

Speaking to reporters midday Thursday, House Majority Leader Eric Cantor (R-Va.) confidently predicted that Plan B would pass. But by mid-afternoon, GOP aides said, it became clear that House leaders had been unable to overcome the skepticism of their members, who were also swayed by veto threats from the White House and Reid’s vow that the measure would die in the Senate.

More than four dozen House Republicans were either on the fence or signaling their opposition heading into the evening vote. Shortly after 7 p.m., the House adjourned and Republican leaders rushed to huddle privately in Boehner’s office just off the Capitol rotunda, where the late senator Daniel K. Inouye (D-Hawaii) was lying in state.

On his way to pay his respects to Inouye, Rep. Tim Scott (R-S.C.) said he had “no clue” what was happening with Plan B. Asked whether he planned to vote for it, he said again: “I have no clue.”

Rank-and-file lawmakers were then summoned to an emergency meeting. In a break with his custom, Boehner offered to open the meeting with a prayer.

“God, grant me the serenity to accept the things I cannot change, the courage to change the things I can, and the wisdom to know the difference,” he said.

Minutes later, Boehner’s office issued a terse statement calling off the vote. And he and other Republican leaders hurried out of the Capitol without answering reporters’ questions.

Felicia Sonmez, Ed O’Keefe and Karen Tumulty contributed to this report.