Donald Trump said Joe Biden "disparaged and insulted" the black community when the former vice president asked an African American journalist if he was a "junkie".
"So I just watched a clip. And Joe Biden, this morning, totally disparaged and insulted the black community. What he said is incredible. And I don't know what's going on with him, but it was a very insulting statement he made," Mr Trump said.
"And I guess you'll figure that out. You'll see it in a little while. But it was a great insult to the black community," the president said.
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"No, I haven't taken a test. Why the hell would I take a test?" the Democratic presidential nominee said.
"Come on, man. That's like saying you — before you got on this program, you take a test where you're taking cocaine or not. What do you think? Huh? Are you a junkie?" Mr Biden said, rhetorically but clearly exasperated.
Mr Trump, who has called himself a "very stable genius" and insisted in a recent interview he is "cognitively there," has openly questioned whether Mr Biden still has the mental capacity to serve as president.
Democrats and the Biden campaign have accused that line of attack as ageist and scoffed at such notions.
"It was a preposterous question deserving of a response that showed the absurdity of it all," a Biden campaign official said of the journalist's question about whether Mr Biden had taken a cognitive test, Politico reported.
The Trump campaign's apparent strategy to paint Democratic Mr Biden as senile and incompetent does not appear to be catching on with voters, a series of Fox News polls has found.
In fact, it's Mr Trump whose mental faculties more voters question.
Forty-seven per cent of the 1,104 registered voters surveyed for the most recent poll from July said they believe Mr Biden has "the mental soundness to serve effectively as president," while 39 per cent said they believe he does not.
Mr Trump scored worse. Just 43 per cent of those surveyed said he is mentally sound enough to serve effectively, while more than half, 51 per cent, said he is not.
Mr Biden, who for decades has been well-known for his verbal gaffes, would be the oldest person to take the presidential oath of office. Mr Trump, who was 70 when he swore the presidential oath in 2017, currently holds that distinction.
Both men's doctors have said they are cognitively high-functioning, according to their campaigns.