“It will be no one thing,” he added. “When you call down the thunder, sometimes it’s not pretty.”

Conservatives say emotions over immigration run so high that the issue could be even more politically potent than the Affordable Care Act. Like many of the economic concerns that animated Tea Party supporters, immigration issues play to people’s anxieties about their financial well-being and the future. Many conservatives who have long mistrusted Mr. Obama because they think his policies will fundamentally alter America believe that his new immigration order will do just that, with millions of potential new foreign-born citizens even though the president’s action does not call for a path to citizenship.

The conundrum for the Republican Party is how to channel that energy. Turned against liberalism, as it was in the 2010 elections that ousted Democrats from power in the House of Representatives, it can deliver serious political advantage. But turned inward, as it so often has been over the last four years, it threatens to tear the party apart. In Virginia, Mr. Cantor’s ouster so emboldened activists that they have started using “to Cantor” as a shorthand for defeating establishment Republicans.

Conservatives see a moment of truth for Tea Party supporters as well. If they think Republican leaders in Congress are not doing enough to fight Mr. Obama on immigration, what is their recourse?

“What the Tea Party has struggled with doing is translating their ideological appeal into political clout,” said Laura Ingraham, the conservative author and radio host. “They don’t have a lot of political clout. They can get out the vote, yes. But I’m talking about getting individual committee chairmen and senators who can mount a real challenge to the establishment forces when required. And can you do that from within the Republican Party?”

One challenge is that some of the party’s biggest financial backers want to see an immigration overhaul pass in Congress. And groups that helped finance the Tea Party’s rise, like Americans for Prosperity, which is supported by Charles and David Koch, will not be there to help the anti-immigration overhaul cause.

Republicans on Capitol Hill are discussing ways to challenge Mr. Obama but are struggling to find a path that does not anger the right. One would be to go to court; another, though it has been discounted by the House Appropriations Committee, would be to try to cut off funding. Other possible pressure points include refusing to confirm the president’s nominees, like Loretta E. Lynch, Mr. Obama’s pick to replace Eric H. Holder Jr. as attorney general.