Earlier on Friday, Trump expressed frustration at the time it was taking the FDA to issue the authorisation.
In a Twitter post, Trump called the FDA a "big, old, slow turtle" and ordered its head to "get the dam [sic] vaccines out now".
A few hours later, The Washington Post reported Trump's White House chief of staff, Mark Meadows, had told FDA Commissioner Stephen Hahn to submit his resignation if the agency failed to issue the authorisation by the end of the day.
Hahn denied the newspaper's reporting.
Initial doses of the Pfizer vaccine are scarce and rationed as the US joins Britain and several other countries in scrambling to vaccinate as many people as possible ahead of a long, grim winter.
It will take months of work to tamp down the coronavirus that has surged to catastrophic levels in recent weeks and already claimed 1.5 million lives globally.
The FDA's authorisation sets off what will be the largest vaccination campaign in US history - but it also has global ramifications because it's a role model to many other countries facing the same decision.
The world desperately needs multiple vaccines for enough to go around, and the Pfizer-BioNTech shot is the first based on rigorous scientific testing to emerge from that worldwide race - a record-setting scientific achievement that shaved years off the usual process.
"I don't think you would have found a scientist on this planet that would have predicted this 11 months ago," said Dr Paul Offit, a vaccine expert at Children's Hospital of Philadelphia who advises the FDA.
The US is considering a second vaccine, made by Moderna, that could roll out in another week. In early January, Johnson & Johnson expects to learn if its vaccine is working in final testing.
Europe is set to make its own decision on the Pfizer-BioNTech and Moderna shots later this month.
China and Russia didn't wait for final-stage tests before beginning vaccinations with some homegrown shots.
Getting shots into arms is the big challenge in the US, especially as a new poll from The Associated Press-NORC Centre for Public Affairs Research found only half of Americans want the vaccine when it's their turn. About a quarter say they won't get it and the rest aren't sure.
