He had visited France, Italy and Los Angeles in the weeks leading up to the debate, but had been at Camp David for a week to recuperate and prepare for the event.
The president told donors after the debate that he “fell asleep on stage” and joked that he should have listened to his team about resting more.
His decision not to work past 8pm comes after a report by Axios, a political news website, that he is only cogent between 10am and 4pm each day.
Although at 81 years old Mr Biden could be expected to move more slowly, some have raised concerns he has a hidden medical condition.
Kevin C O’Connor, the White House physician, said in February the president was “fit for duty” and that tests had turned up no findings which would be consistent with” Parkinson’s disease.
Mr Bates told the New York Times on Thursday that Dr O’Connor had found no reason to re-evaluate Mr Biden for the disease, and that he had not taken Levodopa or other drugs for that condition.
On Thursday, the statistician and writer Nate Silver joined Lawrence K Altman, a clinical professor, in calling for Mr Biden to undergo an “unbiased” assessment of his mental fitness.
Dr Altman at New York University said the public “deserves full reports” from doctors of Mr Biden’s “recent medical examinations”.
Writing on X, formerly Twitter, Mr Silver said: “Since he wants to be president until he’s 86 years old, and a reasonable person might conclude he’s something other than up for the task, how about we get an unbiased checkup from a mutually agreed-upon independent physician?”