WASHINGTON — President Biden and his GOP rival, President Trump, were sweeping the coast-to-coast contests on Super Tuesday, all but cementing a November rematch and increasing pressure on the former president’s last major rival, Governor Haley, to leave the Republican race.

Messrs. Biden and Trump had each won Texas, Alabama, Colorado, Maine, Oklahoma, Virginia, North Carolina, Tennessee, Arkansas, Minnesota, and Massachusetts. Mr. Biden also won the Democratic primaries in Utah, Vermont and Iowa.

Mrs. Haley’s strongest performance was in Vermont, where she was essentially tied with Mrs. Trump in early results. The former president, though, carried other states that might have been favorable to Mrs. Haley such as Virginia and Maine, which have large swaths of moderate voters like those who have backed her in previous primaries.

Not enough states will have voted until later this month for Messrs. Trump or Biden to formally become their parties’ presumptive nominees. The primary’s biggest day, though, made their rematch a near certainty. 

Both the 81-year-old Mr. Biden and the 77-year-old Mr. Trump continue to dominate their parties despite facing questions about age and neither having broad popularity across the general electorate.

The only contest either of them lost Tuesday was the Democratic caucus at American Samoa, a tiny American territory in the South Pacific Ocean. Mr. Biden was defeated by previously unknown candidate Jason Palmer, 51 votes to 40.

Mrs. Haley, who has argued both Messrs. Biden and Trump are too old to return to the White House, was spending election night watching results at the Charleston, South Carolina, area, where she lives. Her campaign website doesn’t list any upcoming events. Still, her aides insisted that the mood at her watch party was “jubilant.”

Mr. Trump’s Mar-a-Lago estate, meanwhile, was packed for a victory party that featured hors d’oeuvres including empanadas and baked brie. 

Among those attending were staff and supporters, including the rapper Forgiato Blow and the former North Carolina congressman, Madison Cawthorn. The crowd erupted as Fox News, playing on screens around the ballroom, announced that the former president had won North Carolina’s GOP primary.

While much of the focus is on the presidential race, there were also important down-ballot contests. The governor’s race took shape in North Carolina, where the Republican lieutenant governor, Mark Robinson, and Democratic attorney general, Josh Stein, will face off in a state that both parties are fiercely contesting ahead of November.

California voters were choosing candidates who will compete to fill the seat long held by Senator Feinstein. And at Los Angeles, a liberal prosecutor attempted to fend off an intense reelection challenge in a contest that could serve as a barometer of the politics of crime.

The earliest either Messrs. Biden or Trump can become his party’s presumptive nominee is March 12 for Mr. Trump and March 19 for Mr. Biden. Both, however, are already signaling publicly that they are looking forward to facing each other again.

“We have to beat Biden — he is the worst president in history,” Mr. Trump said Tuesday on “Fox & Friends.”

Mr. Biden countered with a pair of radio interviews aimed at shoring up his support among Black voters, who helped anchor his 2020 coalition.

“If we lose this election, you’re going to be back with Donald Trump,” Mr. Biden said on the “DeDe in the Morning” show hosted by DeDe McGuire. “The way he talks about, the way he acted, the way he has dealt with the African American community, I think, has been shameful.”

Despite Messrs. Biden’s and Trump’s domination of their parties, polls make it clear that the broader electorate does not want this year’s general election to be identical to the 2020 race. 

A new AP-NORC Center for Public Affairs Research poll finds a majority of Americans don’t think either Messrs. Biden or Trump has the necessary mental acuity for the job.