Speaker Kevin McCarthy’s leadership position is in peril after his most outspoken Republican critic, Representative Matt Gaetz of Florida, said on Sunday he would follow through on threats to try to remove Mr. McCarthy from the job.

Mr. Gaetz’s announcement came a day after Mr. McCarthy, in a stunning reversal, turned to Democrats to help him steer a 45-day stopgap spending bill through the House. The backlash was almost immediate, as Republican hard-liners left the Capitol on Saturday complaining that Mr. McCarthy had sold them out by working with Democrats. They questioned whether he deserved to keep his job.

“I think we need to rip off the Band-Aid,” Mr. Gaetz said during an interview on CNN’s “State of the Union” on Sunday, promising in the next few days to bring up a measure called a “motion to vacate.’’ The move prompts a snap vote on whether to keep the speaker in his post. “I think we need to move on with new leadership that can be trustworthy.”

Mr. Gaetz’s move is the culmination of a monthslong power struggle between Mr. McCarthy and far-right members of his party, who sought to deprive him of the speaker’s gavel in January and shut down the House floor this spring to protest a bipartisan deal Mr. McCarthy struck with President Biden to raise the debt ceiling.

The possibility that Mr. McCarthy might face calls for his ouster has loomed over him since he took the job. Because of a concession he offered to right-wing holdouts in exchange for the speakership, any single lawmaker can call for a removal vote.

But on Sunday, Mr. McCarthy shrugged off the threat, predicting that Mr. Gaetz’s effort to remove him would fail. He said Mr. Gaetz was motivated by a petty grudge rather than a substantive dispute.

“I’ll survive,” Mr. McCarthy said during an interview on CBS’s “Face the Nation.” “You know this is personal with Matt.” He accused Mr. Gaetz of being “more interested in securing TV interviews than doing something.”

“So be it, bring it on,” Mr. McCarthy added. “Let’s get over with it and let’s start governing. If he’s upset because he tried to push us into a shutdown and I made sure the government didn’t shut down, then let’s have that fight.”

For several weeks, Mr. Gaetz has threatened to topple Mr. McCarthy. He complained that the speaker had reneged on several promises he made to Republican hard-liners to win their support to become speaker, including demands for deep spending cuts. In Sunday’s CNN interview, he accused Mr. McCarthy of lying to his G.O.P. members during spending negotiations and making a “secret deal” with Democrats concerning future funding for Ukraine. He and dozens of other conservative Republicans oppose more aid to Ukraine.

“Nobody trusts Kevin McCarthy,” Mr. Gaetz said, predicting that the only way Mr. McCarthy would remain speaker by week’s end is “if Democrats bail him out.”

Though most House Republicans still support keeping Mr. McCarthy on as speaker, Mr. Gaetz’s plans pose an existential threat to his tenure because of the slim majority the G.O.P. holds in the chamber. If Democrats were to vote against Mr. McCarthy — as is almost always the case when a speaker of the opposing party is being elected — Mr. Gaetz would need only a handful of Republicans to join the opposition to remove him, which requires a simple majority vote.

To avoid that fate, at least some Democrats would have to either vote to keep Mr. McCarthy in office, skip the vote or vote “present” — neither for or against. That would lower the threshold for a majority and make it easier to defeat Mr. Gaetz’s motion.

It is not clear whether Democrats would help Mr. McCarthy. They are angry that he recently announced that the House was opening an impeachment inquiry into Mr. Biden, despite lacking evidence of wrongdoing. Most Democrats regard Mr. McCarthy as an untrustworthy figure who has spent months catering to the whims of his right wing.

He has turned to Democrats only when his back is against the wall, as he did in the spring to avoid a federal debt default and again on Saturday to keep the government open.