Donald J. Trump’s victory in the New Hampshire primary on Tuesday provided him the second of an opening pair of wins in the Republican nomination fight that accelerated his push for the party to coalesce behind him and deepened questions about the path forward for Nikki Haley, his lone remaining rival.

The defeat of Ms. Haley in New Hampshire came eight days after the former president trounced Gov. Ron DeSantis of Florida so thoroughly in Iowa that it drove Mr. DeSantis from the race. Mr. Trump and his allies have turned his twin early wins into a milestone — declaring, after just the first two contests, that the party needs to unite behind him now to prepare for a rematch in November between Mr. Trump and President Biden.

No Republican candidate has ever won the first two states and then not ultimately secured the presidential nomination, a fact that Mr. Trump himself noted in his victory speech in Nashua, N.H.

“When you win Iowa and you win New Hampshire, they’ve never had a loss — there’s never been — so we’re not going to be the first, I can tell you,” Mr. Trump told the crowd.

Regardless of what comes next, the win on Tuesday sealed Mr. Trump’s status as the party’s standard-bearer in the history books: Before Mr. Trump, the only Republicans who have ever won both the Iowa caucuses and the New Hampshire primary have been sitting presidents.

The race was called on Tuesday night by The Associated Press the moment the last polls closed, sapping any drama from the outcome. Minutes later, Ms. Haley raced to speak first at her own election party in Concord, N.H., forcefully pressing her case that nominating Mr. Trump would be tantamount to conceding the general election to Democrats.

“You can’t fix the mess if you don’t win an election,” she said. “A Trump nomination is a Biden win and a Kamala Harris presidency.”

Ms. Haley pledged to press forward despite the loss on Tuesday. “New Hampshire is first in the nation — it is not the last in the nation,” she declared. “This race is far from over.”

Before Mr. Trump even took the stage on Tuesday night, the former president called Ms. Haley “delusional” in a social media post, one of several he wrote in all capital letters while she spoke.

It was a preview of a caustic and sometimes crude speech by the former president, in which he used the national platform of a victory address to bash his lone remaining rival, whose voters he would eventually need to win over in the fall.

“She didn’t win. She lost,” Mr. Trump said, calling her an “impostor” that he had beaten “so badly.” He mocked Ms. Haley for delivering an overconfident concession speech: “This is not your typical victory speech, but let’s not let somebody take a victory when she had a very bad night.”

Republicans began almost immediately to ratchet up pressure on Ms. Haley to quit.

“It’s time to drop out,” said Taylor Budowich, the chief executive of Mr. Trump’s super PAC. Senator John Barrasso of Wyoming, who serves in the Republican leadership and had previously endorsed the former president, called Mr. Trump on social media the “presumptive” nominee. And Senator John Cornyn of Texas, who has been critical of Mr. Trump, formally backed him, declaring, “Republicans need to unite around a single candidate.”

Ms. Haley, Mr. Trump’s former United Nations ambassador, had sought for months to narrow the 2024 primary to a one-on-one race with him. She got what she wanted on Sunday with Mr. DeSantis’s exit, giving her only a single full day before voting began in New Hampshire to prosecute her case to independent voters and Republicans that she would be the strongest Republican candidate against Mr. Biden.

In New Hampshire, she did everything she could, from pouring beers to holding babies, as she blitzed across the state alongside its Republican governor, Chris Sununu, who had endorsed her.

But New Hampshire voters appeared to look past Ms. Haley’s warnings that Mr. Trump, who has been indicted four times in the last year and faces 91 felony criminal counts, would bring “chaos” to the campaign trail and be uniquely vulnerable to defeat in a general election.

The attacks between Mr. Trump, 77, and Ms. Haley, 52, had sharply escalated in recent days.

He returned to his nativist playbook to emphasize her birth name, and then purposefully mangled it in social media posts, and even indulged in birther conspiracy theories about her eligibility to serve because she is the daughter of Indian immigrants (she was born in America). Ms. Haley questioned Mr. Trump’s mental acuity after he confused her name with Nancy Pelosi’s, using the incident to press for generational change.

In her concession speech on Tuesday, she cited that verbal slip-up as someone shouted “Geriatric!”

Ms. Haley told the crowd, “The first party to retire its 80-year-old candidate is going to be the party that wins this election.”

Now, Ms. Haley must find traction beyond the first two states, where almost all the campaigning and advertising had occurred. Her super PAC has spent more than $71 million so far — and 99.9 percent of those funds were poured into Iowa or New Hampshire, according to federal records.

Ms. Haley faces what could be an excruciatingly long month. She opted not to compete in the Nevada caucuses with Mr. Trump on Feb. 8 after the state party made rules favorable to him.

“I’m pleased to announce we just won Nevada,” Mr. Trump declared on Tuesday. The formal Nevada caucuses are not for another two weeks, but because Mr. Trump is the lone remaining serious G.O.P. candidate in the running for delegates, he is expected to win all of them.

The next significant clash between Mr. Trump and Ms. Haley will be Feb. 24, in the primary in Ms. Haley’s home state of South Carolina, where she once served as governor.

As the calendar slows, it is Mr. Trump with the political momentum.

In the last 10 days, four of Mr. Trump’s vanquished rivals have all lined up behind him: Mr. DeSantis, Vivek Ramaswamy, Gov. Doug Burgum of North Dakota and South Carolina’s junior senator, Tim Scott, whom Ms. Haley first appointed to the Senate.

“You must really hate her,” Mr. Trump joked to Mr. Scott onstage on Tuesday.

Mr. Scott made his way to the microphone next to Mr. Trump and replied, “I just love you.”

On Monday, Mr. Trump was also endorsed by a Republican lawmaker from Ms. Haley’s home state, Representative Nancy Mace, whom Mr. Trump had tried to oust after she harshly criticized his conduct around the Jan. 6, 2021, riot at the Capitol. Ms. Haley had campaigned with Ms. Mace in 2022.

“You’re seeing this unification moment happening, when in a normal primary you might see it happen in June or July,” Senator J.D. Vance, Republican of Ohio, who campaigned this week for Mr. Trump in New Hampshire, said in a brief interview. “You’re seeing it in January, because the race is effectively over.”

Ms. Haley has pledged to push forward.

“There are dozens of states left to go,” Ms. Haley said on Tuesday night. “And the next one is my sweet state of South Carolina.”

The Haley campaign has already announced a $4 million ad campaign in South Carolina and has fund-raising trips to New York, Florida, California and Texas in the next two weeks to refill her coffers. Ms. Haley’s campaign and her allies have argued that Mr. Trump remains near the 50 percent mark in support in the first two states, a sign of potential vulnerability because as a former president he is universally known.

The strategists leading the Trump campaign, Susie Wiles and Chris LaCivita, predicted in their own memo ahead of the vote on Tuesday that Ms. Haley would be — in all capital letters — “demolished and embarrassed” in South Carolina if she did not quit the race before then.

Scott Reed, a veteran Republican strategist who had worked on a super PAC supporting former Vice President Mike Pence, said Ms. Haley had now ceded her best opportunity to score an outright early win.

“This is a black-and-white business — you either win or you lose,” Mr. Reed said, before invoking the famous rental-car commercial wars of the past. “It’s hard to go on being Avis — ‘We’re number two or number three!’ — behind Hertz.”