Even as conservatives rage against not having 60 votes in the Senate to overcome Democratic filibusters, or the two-thirds majority needed to override a presidential veto, their fiery language is almost certain to diminish the party’s chances of expanding its majorities. That would require winning seats in swing states and districts, where voters often prefer more centrist views.

“A lurch to the right is suicidal,” said Gregory Slayton, a Republican fund-raiser who backed Gov. Scott Walker of Wisconsin in the presidential race before he dropped out a week ago.

One Republican who has stoked the revolt in the House, Senator Ted Cruz of Texas, has vexed Senate Republican leaders and increasingly found himself isolated on policy debates as he has been in advocating a government shutdown again, this time over Planned Parenthood.

But he is getting support for his presidential bid from the party’s more conservative quarters.

“Something needs to change; the writing is on the wall,” said Michael Needham, who heads Heritage Action for America, a conservative advocacy group that has sharply criticized the party leadership. “Are we going to be the party of the donor class or the party of free enterprise and conservative values?”

Representative Bill Flores of Texas, head of the Republican Study Committee, said the hard-liners often seemed bent on destruction. “If you look at what’s happened the last few weeks, you have had people trying to burn the House down,” he said.

One of the first tests of where Republicans are headed will come in the race to succeed Mr. Boehner. Though Mr. McCarthy is the favorite, many in his party say they do not simply want a reprise of Mr. Boehner’s leadership. There are also contests for the other top positions, and the fiercest fight is expected to be for majority leader. Among the main contenders are Steve Scalise of Louisiana, currently No. 3 in the leadership; the conference chairwoman, Cathy McMorris Rodgers of Washington; and the Budget Committee chairman, Tom Price of Georgia. Conservatives are demanding that one of their own be represented, and it is not clear that Ms. McMorris Rodgers or Mr. Price would satisfy them.

Similar battles are expected for party whip, the No. 3 post, and potentially for the conference chair should Ms. McMorris Rodgers leave that position.