At a morning briefing on Thursday, officials said three patients had been moved to the Comfort. After The New York Times published an article with that number, Elizabeth Baker, a spokeswoman for the Navy, said the number had increased to 20 by late in the day. “We’re bringing them on as fast as we can bring them on,” she said.

The next day, on Friday, Ms. Baker said the Comfort would begin screening patients for the coronavirus on site by taking their temperature and giving them a short questionnaire, to relieve the burden on hospitals.

Hospital leaders said they had been exasperated by the delays.

Mr. Dowling said he has had to tear his hospitals apart, retrofitting any unused space, including lobbies and conference rooms, into hospital wards. His facilities now house 2,800 so-called Covid patients, up from 100 on March 20, he said. About 25 percent of those are in serious conditions in intensive care units.

Across the city, hospitals are overrun. Patients have died in hallways before they could even be hooked up to one of the few available ventilators in New York. Doctors and nurses, who have had to use the same protective gear again and again, are getting sick. So many people are dying that the city is running low on body bags.

At the same time, there is not a high volume of noncoronavirus patients. Because most New Yorkers have isolated themselves in their homes, there are fewer injuries from car accidents, gun shots and construction accidents that would require an emergency room visit.

Ultimately, Mr. Dowling and others said, if the Comfort refuses to take Covid patients, there are few patients to send. And given the pernicious spread of the disease in New York City, where nearly 50,000 were infected as of Thursday, dividing patients into those who have it and those who do not is pointless, he said.

The solution, he and others said, was to open the Comfort to patients with Covid-19.

“It’s pretty ridiculous,” he said. “If you’re not going to help us with the people we need help with, what’s the purpose?”