Outsider Donald Trump was elected president in significant part because of his promise to shake up Washington. He’ll soon find that one of the most entrenched forces that object to any change affecting them is the White House press corps. As last week’s meetings with the media showed, a clash is coming.

The clash will go beyond ideology and the media’s dislike of Mr. Trump personally. It will happen because, while the press as an institution is largely in decline throughout the U.S., the White House briefing room is one of...