Even as he opened fire at Mr. Tillerson, Mr. Cohen and others he believes betrayed him, Mr. Trump has been busy orchestrating yet another staff shake-up that may leave more raw feelings in its wake.

On Friday, the president named a replacement for Attorney General Jeff Sessions, whom he fired last month after a withering series of attacks, and he was, once again, seemingly on the edge of forcing out his White House chief of staff, John F. Kelly, with whom he is barely on speaking terms.

The dust up with Mr. Tillerson was remarkable even in Mr. Trump’s world of belligerent confrontations. Just a day earlier, former Secretary of State James A. Baker III, who rubbed Mr. Bush’s feet on his deathbed, cried as he eulogized his friend and former president at a Houston church.

Elsewhere in the same city just hours later, Mr. Tillerson, the former chief executive of Exxon Mobil, took the stage for a benefit for the M. D. Anderson Cancer Center at which he presented a blistering account of the president he served.

“It was challenging for me coming from the disciplined, highly process-oriented Exxon Mobil Corporation to go to work for a man who is pretty undisciplined, doesn’t like to read, doesn’t read briefing reports, doesn’t like to get into the details of a lot of things, but rather just kind of says, ‘Look, this is what I believe,’” Mr. Tillerson said in a discussion with Bob Schieffer of CBS News.

Mr. Trump, he added, kept pressing for actions beyond his authority. “So often, the president would say here’s what I want to do and here’s how I want to do it and I would have to say to him, ‘Mr. President, I understand what you want to do, but you can’t do it that way,’” Mr. Tillerson said. “It violates the law.”

Often, he said, Mr. Trump would get frustrated. “I’d say, ‘Here’s what we can do. We can go back to Congress and get this law changed. And if that’s what you want to do, there’s nothing wrong with that,’” Mr. Tillerson said. “I told him, ‘I’m ready to go up there and fight the fight, if that’s what you want to do.’”