But given a chance to respond at his daily news briefing later Tuesday morning, Mr. Cuomo, a third-term Democrat, held back, repeatedly saying that he did not want to argue with Mr. Trump.

“The president was clearly unhappy,” Mr. Cuomo said, noting the “Mutiny on the Bounty” tweet and other remarks by Mr. Trump, who he said “was clearly spoiling for a fight on this issue.”

But Mr. Cuomo said he would not engage further, saying the coronavirus crisis should be a “no-politics zone.”

“The president will have no fight with me,” he said.

The governor said that he still needed the federal government’s help, particularly as the response begins to shift to restoring economic and social activity.

“In this reopening, we could lose all the progress we made in one week,” he said.

On Tuesday evening, Mr. Trump appeared to reverse course on his claims of absolute authority, saying during a White House news conference that he would allow states to implement their own protocols to reopen the economy.

Addressing Mr. Cuomo directly, he said, “I want him to take his time, do it right and then open New York. I’m not putting any pressure on the governors.”