WASHINGTON — Hostilities between the Republican far right and its typically muted center burst into the open on Tuesday, highlighting deep divisions that could bedevil the party’s leaders if they capture a narrow majority in the House next year.

Initially prompted by the anti-Muslim comments of Representative Lauren Boebert of Colorado, the Republican-on-Republican war of words on Tuesday was remarkably bitter and an indication of a brewing power struggle between an ascendant faction that styles itself after President Donald J. Trump and a quieter one that is pushing back.

First, Representative Marjorie Taylor Greene of Georgia called her freshman colleague Nancy Mace of South Carolina “trash” for condemning Ms. Boebert’s remarks in a television interview.

Ms. Mace then used a series of emojis — a bat, a pile of excrement and a crazy clown — to describe Ms. Greene, then kept up a steady stream of social media attacks, calling her a liar, a grifter and a nut.

Representative Adam Kinzinger, Republican of Illinois, came to Ms. Mace’s defense, calling Ms. Greene “unserious circus barker McSpacelaser” — a reference to a social media post that she once circulated suggesting that wildfires in the West had been started by lasers owned by the Rothschilds, a Jewish banking family.

Mr. Kinzinger added that Representative Kevin McCarthy of California, the Republican leader and would-be speaker who has done nothing to discipline rank-and-file members of his conference for bigoted and violent statements, “continues his silent streak that would make a monk blush.”

Then Representative Matt Gaetz of Florida, an ally of Ms. Greene’s, took to Twitter to amplify an attack by the right-wing provocateur Jack Posobiec denouncing Ms. Mace as a “scam artist” for promoting coronavirus vaccinations on CNN.

The carnival-like behavior would amount to little more than a sideshow if it did not have real implications for midterm campaigns and, possibly, a fractured Republican majority in 2023. Party leaders again chose to remain mum as their backbenchers brawled, and Democrats took full advantage of the spectacle.

“The atmosphere is what it has been and what has been created by the Republican Party over the last 50 years, where they have continued to move down the path of divisiveness, of acrimony, of threats and accusations, which have demeaned the politics of America,” Representative Steny H. Hoyer of Maryland, the majority leader, told reporters.

He again called on Republican leaders to discipline their members, referring to the episode that touched off the hostilities: public comments by Ms. Boebert in which she suggested that Representative Ilhan Omar, Democrat of Minnesota and a Muslim who wears a hijab, could be a suicide bomber and called her a member of the “jihad squad.”

The House’s three Muslim lawmakers — Ms. Omar and Representatives Rashida Tlaib of Michigan and André Carson of Indiana, all Democrats — suggested that their party was looking at options to sanction Ms. Boebert.

“Muslims in this country are proud Americans, hard-working members of our community,” Mr. Carson said. “And we are not anyone’s scapegoat.”

These should be heady days for House Republicans. Off-year elections this month showed real disenchantment with Democratic control of the House, Senate and White House. Redistricting in Republican-controlled state legislatures has given the party a running start to win the four or five seats it needs to control the House, and polling suggests that a narrow plurality of Americans would rather have Republicans in control of Congress. Given the party’s structural advantages on redistricting, access to polls and enthusiasm, that suggests a much broader victory would be at hand if the voting were today.

Michael Steel, a former spokesman for Representative John A. Boehner of Ohio, the former Republican speaker, said the party’s leaders should be working behind the scenes to calm dissent and keep members focused on building a platform and an argument for control.

“The top priority right now should be for everyone in the canoe to have their rifles pointing outward, not at each other,” Mr. Steel said. “And the focus should be on addition, not subtraction. That means keeping all the frogs in the wheelbarrow, even if some of those frogs are pretty ugly.”

Instead, Republicans are stepping on their own message. On Tuesday, CNN unearthed another video of Ms. Boebert from September, when she said she turned to Ms. Omar and referred to the “jihad squad,” again insinuating that she could be a suicide bomber.

Ms. Omar has said that no such confrontation occurred. During a call initiated by Ms. Boebert on Monday — ostensibly to offer contrition — the situation only devolved further, as Ms. Boebert refused to apologize and instead demanded that Ms. Omar publicly ask forgiveness for “anti-American” comments.

Democrats were not the only ones who condemned Ms. Boebert’s behavior. Ms. Mace, a highly regarded newcomer and the first woman to graduate from the Citadel military college, appeared on CNN to say, “I have time after time condemned my colleagues on both sides of the aisle for racist tropes and remarks that I find disgusting, and this is no different than any others.”

Ms. Greene, who like Ms. Boebert is a favorite of Mr. Trump’s, criticized Ms. Mace on social media and on Stephen K. Bannon’s broadcast, “War Room,” and condemned Republican leaders.

“They’re always all over us whenever we say or do anything, but it’s the Nancy Maces that should be called out,” Ms. Greene told Mr. Bannon. She added that she, not Ms. Mace, represented the Republican base, a comment seconded by others on the far right, including Representative Paul Gosar of Arizona.

Representative Peter Meijer, Republican of Michigan, defended Ms. Mace.

“Nancy is a serious legislator who rolls up her sleeves and looks for solutions where they can be found, such as federal cannabis decriminalization, but also digs in and fights when progressives put politics above policy,” Mr. Meijer said. “I can’t think of a single credible thing those attacking her have even tried to accomplish.”

Republican leaders were left pointing fingers at their Democratic counterparts, who they said had also taken no action against members who had crossed lines, whether through anti-Israel comments or exhortations to protesters that they said encouraged violence.

Mr. Hoyer did say that Mr. McCarthy reached out to him to say Ms. Boebert wanted to apologize to Ms. Omar, an overture that Mr. Hoyer said would not end well. He was proved correct.

Mr. McCarthy finds himself in a delicate position. He does not know how large a majority his party might win in November, especially since much of the redistricting has focused on shoring up incumbent advantages than creating more competitive races. A sweeping Republican win would allow him to write off the votes of his party’s fringe.

But if the Republicans claim a narrow majority, Mr. McCarthy would need virtually all of the conference’s votes to claim the speakership, a prize he has sought for nearly a decade. The far right brought down Mr. Boehner in 2015, and Republican divisions over the prospects of Mr. McCarthy’s speakership sunk his last run for the post weeks later.

A handful of members, including Ms. Greene, have been cool to the idea of granting him the gavel should his party claim the majority.

Emily Cochrane contributed reporting.