WASHINGTON—President Trump faces increasing public and private scrutiny from his own party over a series of recent White House moves, as the House impeachment inquiry reduces his margin for error with fellow Republicans and makes him more vulnerable to attacks.

In the past several days, Mr. Trump has been forced to drop plans to host next year’s Group of Seven summit at his Doral golf resort, and a top aide has tried to walk back comments linking Ukraine military aid to an investigation of the president’s political opponents....