Scalise, a McCarthy rival, looks to unite Republicans and take his place.

Republicans delayed an election of the full House that had been planned for midafternoon while the party regrouped.

A week and a day after the abrupt and historic ouster of former Speaker Kevin McCarthy at the hands of a small right-wing bloc, Republicans voted behind closed doors, 113 to 99, to name Mr. Scalise, their second-ranking leader, as his successor. Mr. Scalise turned back a challenge by Representative Jim Jordan of Ohio, the chairman of the Judiciary Committee and a favorite of the hard right.

But he still had to win a majority on the House floor, where chaos reigned the last time Republicans tried to elect a speaker, and it was clear that Mr. Scalise did not have enough votes.

Representative Chip Roy of Texas, the policy director for the ultraconservative House Freedom Caucus who engineered what he called a “power sharing agreement” with Mr. McCarthy during his excruciatingly drawn-out election in January, called for a delay.

“I will not be voting for @SteveScalise,” Mr. Roy wrote on the social media site X, complaining that Mr. Scalise was pushing ahead too swiftly in a move he called “unacceptable & purposeful.”

Mr. Scalise said it was crucial that the House quickly reconstitute itself so it could confront challenges at home and abroad.

“We need to make sure we’re sending a message to people all throughout the world that the House is open and doing the people’s business,” he said in brief remarks after his nomination.

After his slender loss, Mr. Jordan met with Mr. Scalise and offered to nominate him on the House floor, according to a spokesman, but his supporters did not appear to be following his lead.

“I just voted for Jim Jordan for speaker on a private ballot in conference, and I will be voting for Jim Jordan on the House floor,” said Representative Marjorie Taylor Greene, a far-right Republican from Georgia.

“I’m not switching my vote,” said Representative Max Miller, Republican of Ohio. “I’m Jim Jordan all the way.”

The situation underscored the formidable challenge any Republican speaker would face in navigating the same dynamics that made the party ungovernable for Mr. McCarthy. The party’s minuscule majority has empowered a far-right faction that will not tolerate compromise with Democrats who control the Senate and the White House.

The Republican infighting has left the House largely paralyzed since Mr. McCarthy’s ouster. Lawmakers were growing increasingly worried about the impact of continuing to operate without a duly elected speaker, including that the chamber might not be able to support Israel after an invasion by the Palestinian group Hamas that has led to more than 1,000 Israeli deaths and scores of hostages being taken.

Early in the day, Mr. Scalise gained ground after Republicans killed an effort to change their party rules for nominating a candidate for speaker by requiring a public roll call and a higher threshold that would have made it more difficult for him to prevail. It had been billed as an attempt to minimize chaos on the House floor and avoid another situation like what played out in January, when it took Mr. McCarthy 15 rounds of votes to win the speakership.

The nomination of Mr. Scalise, 58, who has been the No. 2 leader in the chamber, was a vote of confidence for a deeply conservative Republican who once described himself, according to a local columnist, as “like David Duke without the baggage.” He represents the Louisiana congressional district that elected Mr. Duke, the former Ku Klux Klan leader, to the State Legislature in 1989. His conservative governmental philosophy, the columnist wrote, was not much different from the one Mr. Duke embraced as a politician, including oppositions to taxes and social safety net programs.

In Congress since 2008, he was diagnosed with blood cancer over the summer and is now undergoing intense treatment, which has prompted him to wear a mask to vote on the House floor and to attend news conferences. And in 2017, during a practice for a congressional baseball game, an anti-Trump extremist shot and seriously wounded Mr. Scalise. He still walks with a limp.

Over the past year, Mr. Scalise has been marginalized by Mr. McCarthy, who has privately described him to colleagues as ineffective, checked out and reluctant to take positions, and cut him out of all major decision making. Their icy relationship made it more difficult for Mr. Scalise to consolidate support in the badly fractured Republican conference.