The president’s aides — some of whom have tired of Mr. DeSantis’s special requests and believe he takes advantage of the relationship with the White House — tried to explain to the president that it would be almost impossible to enforce such a quarantine. Mr. Trump has also resisted a broader national stay-at-home order, which his advisers see as at odds with years of Republican orthodoxy about states’ rights, and unfair to states that are not experiencing major outbreaks.

On Sunday, the White House dropped its earlier, rosier suggestion that stringent social-distancing measures could be lifted by mid-April, and extended national guidelines to combat the coronavirus until April 30. And on Tuesday, Mr. Trump sounded a distinctly somber note, as he acknowledged for the first time that the next two weeks will be “painful” and allowed that predictions of more than 100,000 deaths were realistic possibilities.

Still, when Mr. DeSantis on Monday signed a more limited stay-at-home order for four counties from Key West to Palm Beach, he insisted that it would be needed only through April 15. Because most of the affected cities and counties, like other large jurisdictions in the state, had already enacted their own orders, the governor conceded his own action was of little consequence.

Cellphone location data showed that people in places like Jacksonville and Daytona Beach in northeast Florida, which were not on lockdown, were frequently traveling across county lines. Commissioners in southwest Florida’s Lee County, which likewise had no stay-at-home order, maintained that they did not need one, in part because the governor had not suggested it.

“It seems like he’s listening to the Florida Chamber and Associated Industries and business interests more than he is to medical professionals and health care professionals, which is incredibly disappointing,” Mayor Rick Kriseman of St. Petersburg said before the governor’s latest order. He added that neither the governor nor his staff had called at any point to ask about the city’s needs.

“I may not have agreed on a whole lot of things with Rick Scott when he was governor,” said Mr. Kriseman, a Democrat. “But every time there was a storm approaching the state, I got a phone call from him saying, ‘Just wanted to check in, see how you’re doing, is there anything you need?’” (Mr. Scott, now the state’s junior Republican senator, called him on Saturday.)