Others bit their tongues. When in a July interview Mr. Trump told The Wall Street Journal that Apple CEO Tim Cook had promised to build "three big plants, beautiful plants" in the U.S., Apple declined to comment publicly.
Though Mr. Trump had incorrectly portrayed Apple's U.S. plans, Mr. Cook didn't openly challenge the president because "it would have been a tweet war," according to a person familiar with the company.
Over the months, leaders brooded to their boards and families over whether they should take a stand against the president, according to leaders and corporate advisers.
"Informal conversations among board members often revolved around 'what do you tell your kids?' " said Leslie A. Brun, Merck's lead director and CEO of Sarr Group LLC, an investment holding company.
Some business leaders broke with Mr. Trump after the June decision to withdraw from the Paris climate accord.
Dozens had publicly urged the president to stick with the climate deal, and several, including Dow Chemical Co. head Andrew Liveris and Apple's Mr. Cook made the case directly to the president that the treaty benefited job growth.
Within hours of Mr. Trump's decision, Mr. Iger and Mr. Musk quit the presidential council. Mr. Iger told only a few senior executives at Disney about his decision before announcing it in a tweet, according to a person close to the CEO.
By then, individual members of both main business advisory councils had been discussing disbanding them, given they had met only a few times, according to participants. The discussion picked up steam in July, when Mr. Trump moved to bar transgender people from serving in the military, they said.
More executives grew disenchanted after a group of tech-industry leaders saw few results at a White House meeting in June to brainstorm ways to modernize the government.
"It was becoming patently clear that these meetings were more of a time suck than a productive utilization of their time and resources," said a Washington-based consultant who works with CEOs involved on the councils.
In recent weeks, IBM's Ms. Rometty and Boston Consulting Group head Mr. Lesser asked Mr. Schwarzman, the head of the strategy advisory council, about the future of the group and its effectiveness, given the pressure they were under from employees and the public, said people familiar with the group.
The unrest in Charlottesville, and the president's response, pushed even supporters to make a break.
On Sunday night, Merck's Mr. Frazier called Mr. Brun, the Merck lead director, to tell him that his conscience wouldn't permit him to stay on the manufacturing council, Mr. Brun said. Under Armour Inc.'s CEO Kevin Plank soon followed, along with Intel Corp.'s Brian Krzanich.
After Mr. Trump's Tuesday news conference, in which he apportioned blame equally between hate groups and people protesting them in Charlottesville, other executives said they had no choice but to go.
--Ted Mann, Jonathan Rockoff, Andrew Tangel and Ben Fritz contributed to this article.