Mr. Bannon can still wield influence from outside the West Wing. He believes he can use his perch at Breitbart — which has given a platform to the so-called alt-right, a loose collection of activists, some of whom espouse openly racist and anti-Semitic views — to publicly pressure the president.

And he may still play an insider’s role as a confidant for the president, offering advice and counsel, much like other former advisers who still frequently consult with Mr. Trump. Mr. Bannon had formed a philosophical alliance with Mr. Trump, and they shared an unlikely chemistry.

Mr. Bannon has indicated to people that he does not intend to harm Mr. Trump and he has promised to be somewhat reserved about other administration officials, including Jared Kushner, Mr. Trump’s son-in-law and senior adviser, and his wife, Ivanka Trump, the president’s daughter.

“In many ways I think I can be more effective fighting from the outside for the agenda President Trump ran on. And anyone who stands in our way, we will go to war with,” Mr. Bannon said on Friday.

But his former colleagues in the West Wing are uncertain how long that will last.

Joel Pollak, a Breitbart executive, tweeted after Mr. Bannon’s departure was made public a single word with a hashtag: “#WAR.” Mr. Bannon called reporters to suggest Mr. Pollak had gone too far, but he also acknowledged his own disappointment at departing the White House.

He told The Weekly Standard: “The Trump presidency that we fought for, and won, is over. We still have a huge movement, and we will make something of this Trump presidency. But that presidency is over. It’ll be something else. And there’ll be all kinds of fights, and there’ll be good days and bad days, but that presidency is over.”