Gov. Eric Greitens of Missouri, who is charged with invasion of privacy in connection with an extramarital affair, faced a revolt by week’s end from many fellow Republicans who demanded that he resign and prepared to take steps that could lead to impeachment.

Mr. Greitens, just over a year into his first term, showed no sign of stepping down after his arrest on Thursday, but pressure was mounting and the implications were being felt beyond Missouri.

On Friday, officials from the Republican Governors Association announced that Mr. Greitens was giving up a leadership position with the group. Mr. Greitens abruptly canceled a weekend trip to a meeting of governors in the Washington, D.C. area, where he had planned to visit the Pentagon and give a speech about veterans issues. And Missouri Republicans worried aloud that the scandal would hurt their party’s chances this fall in the battle to defeat Claire McCaskill, a Democrat whose United States Senate seat is seen as one of the most contested in the nation.

Around the state on Friday, lawmakers from both parties urged Mr. Greitens to quit and lamented what they said was a lurid, time-consuming distraction from the business of governing Missouri.