The Wyoming Republican gave an unrepentant final speech warning her colleagues that Donald J. Trump was leading them toward “destruction.” They booed and kicked her out.WASHINGTON — In a remarkable display of loyalty to Donald J. Trump, Republicans moved quickly to purge Representative Liz Cheney of Wyoming from House leadership on Wednesday, voting to oust their No. 3 for repudiating the former president’s election lies and holding him responsible for the deadly Jan. 6 riot at the Capitol.The action, orchestrated by party leaders, came by voice vote during a raucous closed-door meeting on Capitol Hill that lasted just 15 minutes. With the votes stacked against her, Ms. Cheney made a defiant final speech rather than fight the ouster, warning that Republicans would follow Mr. Trump to their “destruction” by silencing dissent and refusing to reject the myth of a stolen election. She drew boos from her colleagues.After months of infighting that erupted after the violent end to his presidency, Republicans’ unceremonious ouster of Ms. Cheney, the scion of one of the nation’s most powerful conservative families, reflected how far the party has strayed from the policy principles and ideological touchstones that once defined it.The vote — and the manner in which it unfolded — also illustrated how Republican Party orthodoxy has come to be defined more by allegiance to a twice-impeached former president who prizes loyalty over all else than by a political agenda or a vision for governing.It came as more than 100 Republicans, including some prominent former elected officials, said they were considering breaking off and creating a third party if the G.O.P. failed to make major changes to extricate itself from Mr. Trump’s stranglehold and what it called “forces of conspiracy, division and despotism.”In an unrepentant last stand minutes before Republicans voted to strip her of her post, Ms. Cheney urged her colleagues not to “let the former president drag us backward,” according to a person familiar with the private comments. She said that if the party wanted a leader who would “enable and spread his destructive lies,” they should vote to remove her, because they had “plenty of others to choose from.”“I will do everything I can to ensure that the former president never again gets anywhere near the Oval Office,” Ms. Cheney told reporters afterward. “We have seen the danger that he continues to provoke with his language. We have seen his lack of commitment and dedication to the Constitution.”Ms. Cheney’s public comments on Wednesday were an unmistakable jab at Representative Kevin McCarthy of California, the minority leader.After the deadly assault on the Capitol by Mr. Trump’s supporters, Mr. McCarthy said the former president had been responsible for the violence and should have quickly called off rioters who were threatening the lives of members of Congress and his own vice president. But he quickly walked back the criticism, which had enraged Mr. Trump, and since then he has arguably done more than anyone else in his party to keep the former president in the fold.Most Republicans are aware of the iron grip Mr. Trump still holds on the party’s voters.Erin Schaff/The New York TimesAfter backing her in a previous leadership challenge, based on her vote to impeach Mr. Trump, Mr. McCarthy had come to see Ms. Cheney as a distraction and political liability in his quest to regain the House majority and become speaker.For her part, Ms. Cheney made it clear that she regarded her ouster as a historic mistake, and intended to continue to be vocal in her criticism. She invited David Hume Kennerly, a former official White House photographer under President Gerald R. Ford with deep ties to her family, to record the day behind the scenes, and defended her stance in a lengthy television interview with NBC less than an hour after the vote.It represented a remarkable arc for the daughter of former Vice President Dick Cheney, a stalwart Republican who became a despised figure among the left for advancing the claim that Iraq had weapons of mass destruction, a falsehood that drove the nation to war.On Wednesday, though, Democrats lavished praise on Ms. Cheney for her refusal to spread a lie, with Speaker Nancy Pelosi calling her “a leader of great courage, patriotism and integrity,” and pointing to her removal as a troubling sign for Republicans.“For the sake of our democracy, reasonable Republicans across the country must take back their party,” Ms. Pelosi said.Behind closed doors in the Congressional Auditorium, a blue-carpeted, wood-paneled hall where Republicans sat in theater-style seats, Ms. Cheney took to the stage on Wednesday morning to make her parting plea, drawing jeers as she warned her colleagues about the consequences of their current course. She ended with a prayer quoting from the Book of John — “Ye shall know the truth, and the truth shall set you free” — and asking God to “help us to remember that democratic systems can fray and suddenly unravel.”“When they do,” she added, “they are gone forever.”Republicans made it clear they were not interested in those reminders.“She who thinks she leads, but has no followers, is only taking a walk,” said Representative Virginia Foxx, Republican of North Carolina, as she made the motion to recall Ms. Cheney, according to a statement her office released afterward. “Liz, I’m afraid you’re a woman who is only taking a walk right now. You have lost your followers.”Republican leaders, who portrayed Ms. Cheney’s removal as a way to unify the party, declined to allow members to register a position on it.When Representative Tom Reed of New York, a moderate who has announced his retirement from Congress, rose to ask whether a recorded vote was allowed, he was told no. Mr. McCarthy had told his colleagues that a voice vote was important to show “unity,” and that it was time to “move forward” and look toward winning back the House, according to a person familiar with the remarks.When the time came, the ayes loudly drowned out the noes. The ouster was so swift that some Republicans were still trickling in to take their seats when Ms. Cheney strode up the center aisle to make her way to a bank of microphones and reporters waiting outside.Mr. McCarthy left without making a public statement, avoiding reporters altogether.At the White House later in the day, speaking to reporters after meeting with President Biden and other congressional leaders on infrastructure, Mr. McCarthy brushed off a question about comments by Representative Elise Stefanik of New York, whom he has anointed to succeed Ms. Cheney. Ms. Stefanik had echoed some of Mr. Trump’s false claims around widespread voter fraud.“I don’t think anybody is questioning the legitimacy of the presidential election,” he said. “I think that is all over with.”Representative Kevin McCarthy, the minority leader, brushed off reporters’ questions about the party’s leadership after a meeting with President Biden at the White House.Doug Mills/The New York TimesRepresentative Jim Jordan, Republican of Ohio, who helped lead the charge against Ms. Cheney, said his case boiled down to a simple idea: “Can’t have a conference chair who recites Democrat talking points.”“It’s time to focus on stopping the Democrats and save the country,” he said.Mr. Trump weighed in Wednesday morning with statements attacking Ms. Cheney and cheering her removal, including one calling her “a poor leader, a major Democrat talking point, a warmonger and a person with absolutely no personality or heart.”Republicans who have styled themselves in his image also gloated openly, including the freshman Representative Madison Cawthorn of North Carolina, who treated Ms. Cheney to a mocking farewell on Twitter: “Na na na na, na na na na, hey hey, goodbye Liz Cheney.”For all the backbiting and turmoil, the episode has also exposed a party unmoored from its traditional policy prerogatives even as Republicans hope to retake control of the House in 2022. Aware of the iron grip Mr. Trump holds on their voters and donors, Mr. McCarthy and other Republicans have made the calculation that to retake the majority, they need the former president’s support — or at the very least, cannot afford to invoke his wrath.“The notion that 5 percent of the Republican Party is going to eviscerate the influence of President Trump in the party never was plausible,” said Representative Dan Bishop, Republican of North Carolina. “It’s not good for the Republican Party. It’s not good for the country.”But the episode has only called attention to the party’s devotion to Mr. Trump, its tolerance for authoritarian tendencies, and internal divisions between Trump acolytes and more traditional Republicans about how to win back the House in 2022. All those dynamics threaten to alienate independent and suburban voters, undercutting what otherwise appears to be a sterling opportunity for Republicans to reclaim the majority.Representative Ken Buck, a conservative Colorado Republican who did not vote to impeach Mr. Trump and voted against removing Ms. Cheney on Wednesday, warned that the party’s rush to purge dissenting voices was more politically dangerous than Republican leaders seemed to understand.“Liz didn’t agree with President Trump’s narrative, and she was canceled,” Mr. Buck said.“It’s time to focus on stopping the Democrats and save the country,” said Representative Jim Jordan, Republican of Ohio, who helped lead the charge against Ms. Cheney.Stefani Reynolds for The New York Times“We have to deal with this narrative at some point,” he continued. “There are major issues — the border, spending. But to suggest that the American people in 2022 won’t consider the fact that we were unwilling to stand up to a narrative that the election was stolen, I think will be taken into consideration with their vote.”Republicans’ choice of replacement for Ms. Cheney has amplified that narrative. While party leaders have cast Ms. Stefanik as a unifying figure who will stick to the party script and stay focused on taking back the majority from Democrats, the New York Republican has resurrected Mr. Trump’s false claims of widespread voter fraud and pledged to stand by him and “his focus on election integrity.”Ms. Stefanik’s hasty ascension has also sparked discontent among the hard-right members of the Republican conference, who are suspicious of her recent transformation from a mainstream moderate who was skeptical of Mr. Trump into one of his most vocal defenders. They have taken issue with her voting record, including voting against his signature 2017 tax cuts bill and his efforts to seize funding to build a border wall.“I think she’s a liberal,” Mr. Buck told reporters on Wednesday. But he added that with top Republicans and Mr. Trump uniting to support Ms. Stefanik, he doubted that anybody would want to risk a future chairmanship or leadership role should Republicans win back the House by taking her on. “Which I think is terribly unfortunate,” he added.Representative Chip Roy, Republican of Texas, circulated a broadside on Tuesday accusing Ms. Stefanik of being insufficiently conservative and said top Republicans were making a mistake by rushing to elevate her. On Wednesday, he appeared to entertain running against her.“She should have an opponent,” Mr. Roy said.A vote on whether to elevate Ms. Stefanik to the No. 3 post is expected on Friday. More