Regeneron, which filed an application with regulators within hours of the president’s video, must now shepherd its antibody treatment through a politically fraught approval process, where the president’s over-the-top endorsement has likely raised the profile of its product, but could also sow suspicion about whether it works.

“I don’t see how it is going to end up being good for a pharma company,” said Ronny Gal, a pharmaceutical analyst for the Wall Street firm Bernstein. “Once you become a political opinion, that’s not great.”

Already, Regeneron is fielding messy questions about how its treatment was tested using cells originally derived from an aborted fetus — a line of research that Mr. Trump has opposed — and the president’s relationship with Regeneron’s chief executive.

Mr. Trump has further complicated the potential rollout of these treatments by pledging — first on Wednesday and again in another video Thursday — that the drugs would be free of charge and would be soon be available in hundreds of thousands of doses.