As he left the floor Tuesday afternoon after Representative Jim Jordan of Ohio was soundly rejected for House speaker, Representative James Comer, Republican of Kentucky and a strong Jordan backer, spoke for many in their embattled party: “I don’t know what to think.”

While it was impossible to determine how, when or even if Republicans might emerge from their ongoing chaos, what was clear was that the House G.O.P. is badly stuck and in crisis, unable to settle on a leader at a time of international upheaval.

Adding to the uncertainty, Mr. Jordan, the hard-right rabble rouser who built his reputation around torpedoing compromises by mainstream Republicans that he found insufficiently conservative, was in the improbable position of having to haggle with those same colleagues to win their support. It was an uphill fight for a man once branded a “legislative terrorist” by a speaker of his own party, and one to which he was not particularly well suited after 16 years in Congress in which he has not sponsored a single bill that has become law.

The dilemma encapsulated the turmoil inside the House G.O.P. two weeks after Representative Kevin McCarthy was ousted from the speakership by a band of far-right rebels who were livid about his compromises with Democrats to head off a catastrophic federal debt default in the spring and a government shutdown this month. Now House Republicans find themselves unable to coalesce around Mr. Jordan, the uncompromising nominee to replace him. Mainstream lawmakers — usually the ones trying to cut deals and reach consensus — have refused to countenance the prospect of Mr. Jordan ascending to the job second in line to the presidency.

Some remain upset at the way Mr. McCarthy was removed. Others are angry that many Republicans abandoned Representative Steve Scalise of Louisiana, the party’s No. 2 in the House as its majority leader, after he initially defeated Mr. Jordan in an internal vote for the speaker nomination. Some just don’t want Mr. Jordan and view him as a threat to their re-elections and a functioning government.

“I voted for the guy who won the election,” said Representative Mario Diaz-Balart, Republican of Florida, who cast his vote for Mr. Scalise on the floor and did not appear inclined to give any ground in the future, no matter how many votes were held.

With Mr. Jordan at least initially stymied and more voting delayed until Wednesday, Republicans and Democrats were intensifying quiet discussions about a potential solution that would somehow empower Representative Patrick T. McHenry, who has been serving as speaker pro tem since Mr. McCarthy was toppled, to conduct the business of the House even temporarily. Some said they increasingly saw it as the only way out, and Mr. Diaz-Balart said it would be a “very prudent and smart move” to look to Mr. McHenry.

Members of both parties were increasingly frustrated as the embarrassing and paralyzing stalemate left the House unable to respond to the conflict in the Middle East or do virtually anything else. The optics worsened Tuesday with the Senate back in town after a recess and moving toward a resolution in support of Israel.

After Republicans pulled the plug on a second speaker vote on Tuesday, Representative Hakeem Jeffries went out to the Capitol steps to castigate Republicans for their predicament. He urged them to work with Democrats to find a solution, though he was not specific what that might be. He suggested he did not expect Republican support to become speaker himself but said that there were “many good men and women on the Republican side of the aisle who are qualified to be the speaker of the House of Representatives” — not including Mr. Jordan, he added, calling him the “poster boy for MAGA extremism.”

It was not a sign of strength on the part of Mr. Jordan and his supporters that after declaring there would be a second vote on Tuesday, they abandoned that strategy, presumably aware that they would lose again. His detractors, on the other hand, had pushed for an immediate reconsideration of the vote, hoping they could defeat him a second time and perhaps push him to step aside as Mr. Scalise did last week when it became clear he could not prevail on the floor.

Mr. Jordan and his allies hoped to work the holdouts through a mix of pleas for party unity, negotiations and a social-media fueled pressure campaign that was already stirring a backlash and could cause more defections in the next round. Mr. Jordan also was apparently rebuffed by Mr. Scalise when he asked for help in converting those still simmering over the treatment of the majority leader.

Even as the arm-twisting proceeded, there were questions about what it would take for Mr. Jordan to withdraw should he fall short a second time, given that he has made his reputation as a partisan brawler willing to take a fight as far as possible. Mr. McCarthy, of course, endured 14 rounds of voting back in January before winning the speakership on the 15th and Mr. Jordan might be reluctant to quickly surrender, particularly if he was picking up support even ever so slightly.

The floor showdown also gave Republicans a taste of what they could expect from Democrats if they did end up going with Mr. Jordan. As Representative Pete Aguilar of California, the No. 3 House Democrat, put forward Mr. Jeffries as the Democratic nominee for speaker, he took the rare step of assailing Mr. Jordan at the same time, providing a template for his party’s political attacks on Republicans who embraced the Ohio Republican. He served notice that every comment and position Mr. Jordan had made or taken would be hung around the neck of House Republicans who backed him — and 200 did so on Tuesday, even as he fell short.

Mr. Aguilar noted Mr. Jordan’s support for a rash of far-right conservative policies on abortion and spending as well as his refusal to certify the 2020 election results after the Jan. 6 assault on the Capitol, calling him an “insurrection inciter.”

“Let’s be clear,” he warned Republicans, “a vote for the gentleman from Ohio is a vote to turn your back on national security. It’s a vote to turn your back on a bipartisan path to fund the government and avoid shutdowns.”

With members of both parties wondering who if anyone could collect the 217 votes necessary to win the gavel, it also put the focus on the difficulties looming for the person who ends up in the job.

Appearing on CNN, Representative Ken Buck, the Colorado Republican who voted for Representative Tom Emmer of Minnesota, the No. 3 Republican, for speaker, said that he didn’t really support him for the job and didn’t like Mr. Emmer.

He later clarified on social media that he was joking and did not intend to inflict the pain of the speakership on Mr. Emmer.

“The office of the speaker is the hardest job in Washington,” Mr. Buck wrote. “I wouldn’t wish that on my good friend.”