Donald Trump’s call to temporarily ban Muslims from entering the U.S. revived a testy split between Republican Party congressional and state leaders and the billionaire’s grass-roots supporters.

In Washington, the nation’s two most powerful Republicans, House Speaker Paul Ryan and Senate Majority Leader Mitch McConnell, joined on Tuesday in the GOP establishment’s near-universal condemnation of the presidential candidate’s proposal. But in Iowa and New Hampshire, the key early nominating states, Trump’s backers moved forcefully to counter party insiders’ censure.

“It is un-Republican, it is unconstitutional and it is un-American,” Jennifer Horn, the New Hampshire state party chairman said in a statement.

That prompted New Hampshire state Rep. Al Baldasaro, who backs Mr. Trump, to call for Ms. Horn’s resignation. He said establishment Republicans “don’t like Trump because he tells it like it is. I’m not prejudiced against Muslims, but until we can straighten them out and know who’s who and who is coming into our country, we have to stop the immigration.”

The internal strife, which Mr. Trump has both helped to create and stoked in his quest for the GOP presidential nomination, also rekindled an old threat: That he’ll quit the race and run as an independent candidate in November, significantly upping the chances of a Democrat winning.