The video, which the US President shared to Twitter, also promoted hydroxychloroquine, a drug which Mr Trump regularly touts and has even claimed to have used himself.
Studies have found no evidence that hydroxychloroquine, when used with or without the antibiotic azithromycin - as promoted many times by Mr Trump - helps treat coronavirus infection or prevents people falling seriously ill.
They include studies commissioned by the US National Institutes of Health, the World Health Organisation and universities in the US and abroad.
Because of the lack of benefit and risks of serious side effects such as heart rhythm problems, the US Food and Drug Administration (FDA) recently revoked its brief authorisation of emergency use of the drug for Covid-19.
NIH treatment guidelines also specifically recommend against hydroxychloroquine's use, except in formal studies.
Dr Anthony Fauci, the director of the National Institute of Allergy and Infectious Diseases, told BBC News on Wednesday that Mr Trump's comments were "not helpful".
"This issue of tweeting and retweeting is something that I've never understood or gotten involved in," he said.
"But I can tell you, you'd have to say it's not helpful if people get signals about not wearing masks when we are trying to get people to universally wear masks.
"We know that every single good study - and by good study I mean randomised control study in which the data are firm and believable - has shown that hydroxychloroquine is not effective in the treatment of Covid-19."
It comes after Mr Trump defended the drug at Tuesday's White House briefing.
"Many doctors think it's extremely good and some people don't," he said.
"Some people I think it's become very political. I happen to believe in it. I would take it. I ... took it for a 14 day period. And I'm here. Right?"
In addition to sharing the video, Mr Trump retweeted several posts attacking the credibility of Dr Fauci.
Later, Mr Trump appeared to back away from his criticism of Dr Fauci, saying, "I get along with him very well".
"He's got a very good approval rating, and I like that," Mr Trump added.
He said Dr Fauci and White House coronavirus task force coordinator Dr Deborah Birx worked for him, adding: "And yet they're highly thought of but nobody likes me".
Dr Fauci has become an off-and-on target of Mr Trump and some of his White House aides and outside allies, who disagreed with the doctor's early recommendation to shut down the economy as a way to slow the virus.
Mr Trump, in recent interviews, has described Dr Fauci as "a bit of an alarmist" and accused him of making "mistakes" in his coronavirus guidance.
Dr Fauci on Tuesday said he would deal with the attacks by keeping his head down and doing his job. He also backed the conclusions of the FDA and others about hydroxychloroquine and Covid-19.
Asked if he can do his job while Mr Trump publicly questioned his credibility, Dr Fauci said the stakes were too high not to stay involved.
"We're in the middle of a crisis with regard to an epidemic, a pandemic. This is what I do," Dr Fauci said on ABC's Good Morning America. "This is what I've been trained for my entire professional life and I'll continue to do it."
More than four million people in the US have been infected by the coronavirus and the death toll is nearing 150,000, according to a tally by Johns Hopkins University.