Mr. Biden, speaking at a University of Miami rally to combat sexual assault, said, “A guy who ended up becoming our national leader said, ‘I can grab a woman anywhere and she likes it,’ ” referring to an Access Hollywood audio recording in which Mr. Trump is heard boasting about kissing and groping women without their consent. Mr. Biden said when he was asked if he would like to debate Mr. Trump, he said, “If we were in high school, I’d take him behind the gym and beat the hell out of him.”

The back-and-forth blustering between two men in their 70s comes a day after Mr. Trump criticized two of his predecessors, Presidents Barack Obama and George W. Bush, for not being able to improve relations with Russia. And Mr. Trump is facing revived sexual misconduct accusations after a New York state judge ruled that a defamation lawsuit from a woman who has said Mr. Trump made unwanted sexual advances could go forward.