HOUSTON—The third Democratic presidential debate left the candidates differing on whether it is fair game to discuss former Vice President Joe Biden’s fitness to be the party’s presidential nominee next year.

Mr. Biden has maintained his front-runner status in public polling despite a series of verbal missteps during the summer. Some Democrats have warned that his misstatements and gaffes could undermine Mr. Biden’s case that he is the one best placed to beat President


And at least one of his lesser-known Democratic opponents,

Rep. Tim Ryan

of Ohio, recently said the 76-year-old Mr. Biden is “declining.”

That sentiment seeped into the Thursday’s debate when

Julián Castro,

the former Housing and Urban Development Secretary, took a shot at Mr. Biden’s memory.

Mr. Biden brushed aside those concerns Friday. Asked by a reporter if he would release his medical records to address concerns about his age and mental sharpness, Mr. Biden responded: “What the hell concerns, man? You wanna wrestle?”

He vowed to release his medical records after his next physical exam and “before there’s a first vote.”

In the debate, Mr. Castro claimed Mr. Biden said contradictory things about whether his health-care plan, which would establish a government-run insurance option, would automatically enroll people.

“Are you forgetting already what you said just two minutes ago? I mean, I can’t believe that you said two minutes ago that they had to buy in and now you’re saying they don’t have to buy in,” said Mr. Castro, who is 44. “You’re forgetting that.”

Mr. Castro apparently was mischaracterizing Mr. Biden’s remarks, according to a transcript of the debate.

After the debate, Mr. Biden’s campaign aides called Mr. Castro’s remark a “cheap shot.”

A few of the Democratic presidential candidates had big moments during the third debate. WSJ’s Gerald F. Seib looks at four who stood out. Photo composite: Associated Press / Getty Images

However, Mr. Biden told reporters Friday that he thought the debate was generally a clean fight.

“I don’t view it as anything. I think he’s got his facts wrong,” Mr. Biden said. Asked if it was fair for his rivals to note his age, he responded: “Sure it is,” and added that he enjoyed the vigorous debate on health care. “What I saw last night is fewer and fewer personal attacks,” he said. “So I think we’re getting closer and closer to what I think everybody’s looking for in the Democratic Party and the country: What is our view of the future, who has the best view and who can best initiate it, who can best complete it.”

He spoke with dozens of black students at Texas Southern University Friday, surprising those gathered near the student center’s Chick-fil-A with hand shakes and selfies.

Earlier in the day, Mr. Biden spoke at a private fundraiser in the affluent River Oaks section of Houston. “Last night, I thought, was a good night. I think I could have done better,” he told attendees. “I will do better, God willing.”

Mr. Castro defended his comments and said they weren’t related to Mr. Biden’s age or mental sharpness. “You know, this is what the media does,” he said. “This was not a conversation about personalities. This was a conversation about health-care policy.”

Mr. Castro added: “When we get on the stage in October 2020 against Donald Trump, does anybody think that he’s going to be the nicest guy in the world?”

After the debate as candidates strolled through the “spin room” of media at Texas Southern University’s campus in Houston, several candidates were asked about Mr. Biden’s mental sharpness and Mr. Castro’s remarks.

“I think that we are at a tough point right now, because there’s a lot of people concerned about Joe Biden’s ability to carry the ball all the way across the end line without fumbling,”

Sen. Cory Booker

(D., N.J.) said on CNN after the debate., adding that Mr. Castro “has every right to call that out.”

“I think that Castro had some really legitimate concerns about, can [Mr. Biden] be someone in a long grueling campaign who can get the ball over the line?” the senator said.

Mr. Booker, 50, said he didn’t think Mr. Biden is too old to be president, but added: “There are definitely moments when you listen to Joe Biden and you just wonder.”

Another presidential contender,

Sen. Amy Klobuchar

(D., Minn.), said on CNN that Mr. Castro was out of line. “I just thought: This is not cool,” she said. “I thought that was so personal and so unnecessary.”

Some of the presidential candidates, including Sens. Elizabeth Warren of Massachusetts and Kamala Harris of California, declined to weigh in.

Friday morning in a television appearance, former Texas Rep. Beto O’Rourke said of his fellow Texan’s comment toward Mr. Biden: “That kind of personal attack I don’t think is what we need right now.”

During the debate, Mr. Biden clashed sharply with Sens. Warren and Bernie Sanders of Vermont about the role of government in the nation’s health-care system, as the three front-runners in the polls faced off for the first time on a debate stage.

In addition to the jab about memory, Mr. Castro also took a shot at Mr. Biden over immigration.

Mr. Biden defended President Obama over his administration’s deportation of undocumented immigrants and deflected questions about his own personal responsibility for the policies.

“He wants to take credit for Obama’s work but not have to answer any questions,” said Mr. Castro, who also had been part of the administration. “I’m fulfilling the legacy of

Barack Obama

and you’re not,” Mr. Castro told Mr. Biden.

The former vice president shot back: “That’ll be a surprise to him.”

Write to Joshua Jamerson at joshua.jamerson@wsj.com

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