The drug maker Regeneron said on Wednesday evening that it had submitted an application to the Food and Drug Administration for emergency approval of the experimental antibody cocktail that President Trump had praised just hours earlier without evidence as a “cure” for the coronavirus.

The company said that at first, access to the treatment would be extremely limited, with only enough doses for 50,000 patients, a far cry from the “hundreds of thousands” of doses that Mr. Trump said in a video released Wednesday he would soon be making available to Americans free of charge.

In the five-minute video, Mr. Trump said that it was a “blessing from God” that he had been infected with the coronavirus and that the Regeneron cocktail had suddenly made him feel better. “I felt unbelievable,” he said. “I felt good immediately.”

There is no evidence that the treatment is the reason he was feeling better, and his doctors have said he has taken other drugs as well.

“I call that a cure,” Mr. Trump said in the video, adding that he would make sure it was in every hospital as soon as possible.

Mr. Trump gave the impression that he would push the F.D.A. to approve Regeneron’s treatment, even though the agency’s scientists are supposed to make independent decisions about approvals.

“I have emergency use authorization all set, and we’ve got to get it signed now,” Mr. Trump said.

A spokeswoman for the F.D.A. declined to comment on Wednesday, saying the agency does not confirm or deny product applications.

The news of Regeneron’s application on the same day that Mr. Trump effusively praised the unproven drug is likely to intensify fears that the president is pressuring federal health agencies to make decisions aimed at benefiting him politically. In the video, Mr. Trump repeated his desire to get a vaccine approved before the election, even though the vaccine makers themselves have said that is highly unlikely.

Regeneron’s treatment is a cocktail of two powerful antibodies that are believed to boost the immune response to the virus. Early results seem promising. In a press release, the company has said the cocktail lowered virus levels particularly in people who had not made their own antibodies, but the company has not yet released detailed data to back up its claim. Clinical trials are not yet complete.

Drugs are not generally considered to be proven safe or effective until they undergo rigorous clinical trials that compare one group of people who received the treatment with those who receive a placebo.

The company has received more than $ 500 million from the federal government to develop and manufacture its experimental treatment as part of Operation Warp Speed, the federal effort to come up with viable vaccines and treatments for the virus and to help distribute them once they are available.

The company said that it expects to have doses available for 300,000 patients in the next few months. Under the agreement with the federal government, it said those doses would be made available free of charge. In August, it announced a partnership with the pharmaceutical company Roche to ramp up production to about 250,000 doses a month by next year.

Mr. Trump received the antibody cocktail on Friday, but it is not the only drug that he was prescribed. He has been taking the antiviral drug remdesivir, as well as the steroid dexamethasone, which the World Health Organization and National Institutes of Health recommend only for people who have severe or critical cases of Covid-19.

In an interview on Wednesday before the company’s announcement, Dr. George Yancopoulos, Regeneron’s president and chief scientific officer, said it was possible that Mr. Trump responded to the treatment and that the level of virus had declined. “That’s a logical conclusion,” Dr. Yancopoulos said. “Based on his symptomology, that has to have happened.”

But neither Dr. Yancopoulos nor Mr. Trump can definitively say whether the treatment worked. Any number of factors could complicate the picture, including the fact that most people who are infected with the virus recover. That is why drugs are typically tested in large clinical trials with hundreds and sometimes thousands of people.

Dr. Peter Chin-Hong, an infectious disease specialist at U.C.S.F. Health in San Francisco, said in his opinion, there was “one million percent no” chance that the Regeneron treatment could have cured Mr. Trump in 24 hours, as the president claimed.

Another explanation, he said, is that the president is experiencing the effects of the steroid dexamethasone, which he has been receiving since Saturday, which is known to reduce fever and can create feelings of well-being and euphoria in patients. “This is all in keeping with the dexamethasone speaking,” Dr. Chin-Hong said.

Gina Kolata contributed reporting.