Referring to the network as “the new Fox News,” he continued by saying the network “is letting millions of GREAT people down!”

“We have to start looking for a new News Outlet,” he added. “Fox isn’t working for us anymore!”

This was not the first time that Mr. Trump had expressed frustration with Fox News, but it was eye-opening. The president has good relationships with many of Fox’s on-air personalities, including the hosts of “Fox & Friends,” Sean Hannity, Jeanine Pirro and Laura Ingraham. His former spokeswoman, Sarah Huckabee Sanders, signed on as a Fox News contributor last week.

But whether this is a sign of genuine deterioration in relations between the president and his preferred cable channel or a short-lived dust-up will be closely watched by news executives and throughout the Beltway.

On Thursday, two on-air personalities responded to Mr. Trump’s tweets.

Neil Cavuto, an anchor for Fox News and Fox Business Network who has been critical of Mr. Trump in the past, said on the air: “Mr. President, we don’t work for you. I don’t work for you. My job is to cover you, not fawn over you or rip you. Just report on you — to call balls and strikes on you. My job, Mr. President — our job here — is to keep score, not settle scores.”

“You’re entitled to your point of view, Mr. President,” he continued. “But you’re not entitled to your own set of facts.”