“It didn’t show a lot of class,” Mr. Manchin said. “If it had been George W. Bush or Reagan or Clinton or whoever, protocol is protocol.”

The invitation proved to be opportune for Mr. Obama, who had been making steady progress in persuading Democrats to delay a vote on sanctions to give him some diplomatic breathing room.

The president had been “changing minds,” said Senator Chris Murphy, Democrat of Connecticut.

“I don’t think the invitation has been helpful to the debate in Congress,” Mr. Murphy added, saying he believed Mr. Netanyahu’s speech would only further politicize an issue that should be above partisanship. “My worry all along has been that Republicans are going to oppose this deal simply because it’s President Obama’s deal, and the invitation to Netanyahu confirms that there are some Republicans who simply put politics ahead of what’s best for the country.”

Other Democrats were also quickly lining up behind the president.