“He’s an important part of our team, and he’s a particularly important part of the budget debate, which will be on the floor next week,” Mr. McConnell said pointedly at an event in Hazard, Ky., with Scott Pruitt, the head of the Environmental Protection Agency.

“Bob Corker has been a leader in Congress on issues as diverse as deficit reduction and combating terrorism, and he is a man of unwavering integrity,” Senator Rob Portman, Republican of Ohio, said in a statement. “If we’re going to accomplish our economic and national security agenda we’re going to have to work together, period.”

Mr. McConnell’s associates were blunter in expressing the leadership’s frustration with the president for disrupting party unity as Republicans push for tax cuts. “This was going to be hard no matter what,” said Billy Piper, Mr. McConnell’s former chief of staff. “And he took the guy who was one of the linchpins of this and incinerated him for no reason.”

The White House spent Monday telling allies that Mr. Corker was responsible for the fight, not Mr. Trump, and that the senator was an attention-seeking obstructionist.

“I find tweets like this to be incredibly irresponsible,” Kellyanne Conway, the president’s counselor, told Fox News, referring to Mr. Corker’s posted response to Mr. Trump on Sunday. “It adds to the insulting that the mainstream media and the president’s detractors — almost a year after this election, they still can’t accept the election results. It adds to their ability and their cover to speak about the president of the United States in ways that no president should be talked about.”

Vice President Mike Pence was left to defend Mr. Trump against what he called “empty rhetoric and baseless attacks,” saying the president had accomplished a lot internationally. “Today our nation once again stands without apology as leader of the free world,” Mr. Pence said in a statement. “That’s what American leadership on the world stage looks like and no amount of criticism at home can diminish those results.”

Mr. Trump has grown frustrated by Senate Republicans — including lashing out at Mr. McConnell for not getting the job done — as legislation to repeal President Barack Obama’s health care program has been repeatedly blocked. He has engaged in open conflicts with Senators John McCain and Jeff Flake of Arizona, Lindsey Graham of South Carolina and Lisa Murkowski of Alaska, among others, although just Monday he went golfing with Mr. Graham.