The shock waves from Mr. Cantor’s defeat by the Tea Party candidate David Brat — and his quick exit from the leadership stage — will have broad ramifications for the remainder of President Obama’s term and the Republican Party’s efforts to regain the White House. Conservative Republicans say their districts are seething with anger over what Tea Party voters see as weak leadership in the fight against an “imperial presidency.”

“That’s what we’re seeing around America. I know in my district people are upset. People want government to be accountable, to start doing what they were sent up to do,” said Representative Ted Yoho of Florida, who in 2012 defeated an entrenched but conservative Republican incumbent.

In the meeting on Wednesday afternoon, Mr. Boehner sought to put to rest any talk of a brush fire that could sweep into the speaker’s office, declaring that he would seek re-election as speaker in the next Congress. Supporters had encouraged him to declare now before the House’s most conservative Republicans became too emboldened.

“With shake-ups like this, it’s really important to have a steady hand on top,” said Representative Devin Nunes, Republican of California. “You can’t lose both of those guys at once,” he added about Mr. Boehner and Mr. Cantor.

Some conservatives indicated they may not be happy unless Mr. Boehner has a scare.

“I’m leaving an open mind to that question,” Mr. Fleming said. “I think undoubtedly everything will be up for discussion.”