Mr Biden is trailing Trump in all seven of the major swing states, and the pair are neck-and-neck in national polls.
The debate on Thursday was considered a major opportunity to spread his message to around 70 million voters, but fell flat after he was unable to perform as planned.
Trump, speaking at his own rally in Virginia on Friday, said Mr Biden “didn’t know what the hell he was doing”.
“The question every voter should be asking themselves today is not whether Joe Biden can survive a 90 minute debate performance, but whether America could survive four more years of crooked Joe Biden in the White House,” he said.
Mr Biden’s allies, including Ms Harris and Mr Newsom, have said he should not stand down and claimed he is the most likely Democratic candidate to win the election.
If Mr Biden was to be replaced, he must choose to hold an open convention in Chicago in August, allowing delegates who pledged to support him after the Democratic primaries to change their vote.