One Trump supporter told the president that he was concerned about nasty messaging in American politics, while another said her family members would not speak to her because of her support for him. She asked Mr. Trump how he would unite the country.

In each case, the president blamed his opponents for incivility. “When they hit us, we have to hit back,” he said. “I wouldn’t be sitting up here if I turned my cheek.”

Mr. Trump defended his handling of the spreading coronavirus, and deflected a question about why he did not act sooner to change Obama-era regulations that he has blamed for delaying testing kits. He said he was “thinking about a lot of other things, too, like trade and millions of other things.”

“As soon as we found out it was a problem,” he added, “we did it.”

Asked whether he would stop shaking hands because of the outbreak, Mr. Trump, a self-described germophobe who has said he envies the Japanese custom of bowing, called it a practical impossibility.

“I love the people of this country, and you can’t be a politician and not shake hands,” the president said. He cast aside the advice of public health officials who have urged Americans to avoid contact, which can spread the coronavirus.

“Now the concept of shaking hands since this — you’re hearing a lot of stuff about trying not to shake hands, and that has not stopped me at all,” he said.

After the event, Mr. Trump stayed for several minutes and shook dozens of hands with no evident distress.