HOUSTON — The long-awaited clash between former Vice President Joseph R. Biden and his far-left rivals finally occurred Thursday on the debate stage, with the two sides exchanging sharp jabs over health care.

Mr. Biden, the front-runner in the race, also found himself under fire from other moderate candidates aiming to dent his seemingly impenetrable armor.

Former Housing and Urban Development Secretary Julian Castro questioned the health and mental vigor of the 76-year-old Mr. Biden.

When Mr. Biden said people would not have to buy into his health care plan, Mr. Castro pounced: “You said two minutes ago that they would have to buy in. Are you forgetting what you said two minutes ago?”

The crowd responded with moans and murmurs.

Mr. Biden, whose health and age have become the subject of much recent speculation, appeared shocked, saying Mr. Castro had his facts wrong.

The fireworks on the stage at Texas Southern University marked a new phase in the contest where it is now or never for candidates to make their mark.

Mr. Biden slammed the proposal by Sen. Bernard Sanders of Vermont and Sen. Elizabeth Warren of Massachusetts for a “Medicare for All” government takeover of health care. He said it would knock nearly 150 million people off their private health insurance plans and result in tax increases for the middle class.

“The senator says she is for Bernie. Well, I am for Barack,” said Mr. Biden, simultaneously bucking Ms. Warren and Mr. Sanders while reinforcing his ties to former President Barack Obama.

The single-payer plan, which would eliminate private health insurance companies, has become a flash point in the fight over the direction of the Democratic Party. Mr. Biden is offering a return to normalcy, while Mr. Sanders and Ms. Warren want major structural changes to the federal government.

“Maybe you run into people who love their premiums. I haven’t,” said Mr. Sanders, who has described his plan as a socialist-style European model.

“We’re in America,” Mr. Biden quipped.

One area of agreement in the debate was the disdain for President Trump and the urgent need to oust him in November 2020.

Several candidates called the president racist and anti-immigrant. Former Rep. Beto O’Rourke of Texas said the gunman who killed 22 people in Texas was “inspired to kill by our president,” and Sen. Kamala D. Harris said Mr. Trump “didn’t pull the trigger, but certainly he has been tweeting out the ammunition.”

They faulted his trade policies and use of tariffs against China with hurting the economy and especially farmers. However, the Democratic hopefuls shied away from promising to lift the tariffs the president has slapped on Chinese goods.

On the immigration issue, Mr. Castro again took a shot at Mr. Biden. When Mr. Biden was pressed to condemn Mr. Obama’s policies that resulted in deportations of about 2.9 million people.

Mr. Biden explained that it was Mr. Obama’s policy.

“I was vice president,” he said.

Mr. Castro pounced again: “He wants to take credit for Obama’s work but not answer any questions. I don’t get that,” he said.

The attacks exposed a split among the candidates who are trying to cast themselves as unifiers and generally speaking have tried to stay above the fray.

“This is why presidential debates are becoming unwatchable,” said Mayor Pete Buttigieg of South Bend, Indiana. “This reminds everybody of what they cannot stand about Washington.”

Mr. Castro fired back with a grin. “That’s an election. This is what we are here for,” he said.

Sen. Amy Klobuchar of Minnesota came to Mr. Buttigieg’s defense by saying, “A house divided cannot stand.”

In the health care fight, the former vice president touted his plan to build on the accomplishments of Obamacare, which expanded the Medicaid program for the poor and provided federal subsidies for people to buy private insurance.

Medicare for All is “a good plan if you like it. I don’t like it,” Mr. Biden said.

He delivered the attack while flanked on stage by Ms. Warren and Mr. Sanders, who took turns pushing back against Mr. Biden.

Ms. Warren extended an olive branch to the Obama legacy by saying, “We all owe a huge debt to President Obama.”

But while acknowledging that Obamacare was a major turning point that extended health care coverage to millions of Americans, she said the next step needs to be universal coverage under Medicare for All.

Others joined in the health care fight.

Ms. Klobuchar, a more moderate candidate who is striving to be an alternative to Mr. Biden, took aim at the Sanders/Warren approach.

“I read the bill,” she said, to counter Mr. Sanders‘ standard line that he “wrote the damn bill.”

“It says we will no longer have private insurance as we know it, and that means that 149 million Americans will no longer be able to have their current insurance,” she said, citing Page 8 as the place in Mr. Sanders‘ bill. “That is in four years. I don’t think that’s a bold idea; I think it’s a bad idea.”

Sen. Cory A. Booker of New Jersey, another more moderate candidate looking to replace Mr. Biden, said pursuing a socialist-style agenda would risk a reelection of Mr. Trump.

“We cannot sacrifice progress on the altar of purity,” Mr. Booker said. “We have one shot to make Donald Trump a one-term president, and we cannot lose it by the way we talk about each other and demonize and degrade each other. We can walk and chew gum at the same time.”

But on guns, candidates wrangled to be the most contrary to the National Rifle Association — Mr. Sanders bragged about his F rating — and gun-rights groups.

Mr. O’Rourke, who used to represent the El Paso area hit in the Walmart massacre, doubled down with gusto on his proposal to require people who own assault rifles to sell them under threat of government force.

“Hell yes. We are going to take your AR-15, your AK-47, we are not going to allow it to be used against our fellow Americans anymore,” Mr. O’Rourke said.

Democrats chose Houston as a debate site to send a clear message that they plan on competing for the states’ 38 electoral votes next fall.

Jimmy Carter in 1976 was the last Democrat to carry Texas in a presidential election, but party officials remain bullish about their chances in 2020. They say the Lone Star State’s changing demographics and rapidly growing Hispanic population could put it in play and help them win down-ticket races.

The three-hour debate, co-hosted by ABC and Univision at Texas Southern University, was held as the shape of the race comes into sharper focus.

Mr. Biden has led in the polls since launching his campaign in April despite a series of verbal miscues, concerns over his age and blowback over stances during his long record of public service that don’t conform to today’s left-wing sensibilities.

Ms. Warren has been on the rise, while Mr. Sanders‘ support appears to have leveled off, raising questions over whether he can expand his base of support.

The three candidates have helped frame the battle between the more centrist and liberal elements of the Democratic Party over whether their 2020 standard-bearer should preach a return to pre-Trump normalcy or promise big structural changes to the federal government.

Thursday marked the first time that Mr. Biden and Ms. Warren have shared a debate stage.

⦁ Seth McLaughlin reported from Washington.

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