What some conservatives aren't willing to do to kill PPACA

'Far more harm than good'

Dan Holler, a spokesman for Heritage Action For America, which along with the Club pushed the unsuccessful legislative strategy to defund Obamacare in 2013, similarly declined to support calls for nixing the filibuster.

"We need a president willing to fight to repeal Obamacare and reform the healthcare system the right way. Fortunately, there is a legislative process already in place: reconciliation," Holler said, referring to the parliamentary tool that lets the Senate bypass the 60-vote threshold to make certain changes to the federal budget.

Ken Cuccinelli, the president of the Senate Conservatives Fund, another well-financed right-leaning group, also poured cold water on the idea.

"Eliminating the legislative filibuster would do far more harm than good because the Senate, despite being controlled by Republicans, is a liberal body that left unrestricted by debate will pass even more liberal legislation," Cuccinelli told Bloomberg in a statement. "The Republicans can use budget reconciliation to repeal Obamacare with a majority vote, just as the Democrats used it to pass Obamacare with a majority vote."

Reconciliation: plausible or 'nonsense'?

That reconciliation strategy has been endorsed by other conservatives including Cruz and Erick Erickson, the editor of the blog RedState. It's a complex scenario involving the use of congressional budget-writing authority—and rulings by the Senate parliamentarian—to allow congressional Republicans to do a back-door repeal of some elements of Obamacare this year. But congressional experts are doubtful about whether it will fly.

"That is, or should be, nonsense. Reconciliation can't be used to increase deficits, and is quite constrained when it comes to substantive measures that are not directly fiscally related," said Norm Ornstein, a congressional scholar at the American Enterprise Institute. "But what this suggests is that right wing groups will try to bludgeon [Senate Majority Leader Mitch] McConnell and [House Speaker John] Boehner to stretch the rules beyond recognition."

The Club for Growth's Roth is floating another proposition, suggesting that a future Republican president could roll back the IRS rule that allows the government to provide PPACA health insurance premium tax credit subsidies through the PPACA exchanges established by the U.S. Department of Health and Human Services (HHS). But the Supreme Court made clear in its 6-3 decision upholding the subsidies that the law doesn't delegate that task to the agency, meaning the next administration cannot reverse it by executive order.

For now, the best Republicans can hope for is symbolic votes to repeal the law and—potentially—passing smaller bills to nix non-essential components of it.

"We will be voting on repeal of Obamacare," said Senate Majority Whip John Cornyn, Republican of Texas, after the Supreme Court ruling in King v. Burwell. Congressional Republicans have approved a non-binding budget resolution calling for repeal of the law, but in the unlikely event such a measure landed on Obama's desk, it would face certain veto. Meanwhile, Republicans plan to push legislation to nix smaller parts of PPACA like the excise tax on the sale of certain medical devices and the cost-cutting Independent Payment Advisory Board, which would only kick in if Medicare spending rises above a certain level.

'A moment of relief' for the White House

Republican presidential candidates are united behind the idea of repealing PPACA. But their task is complicated by their inability to agree on an alternative plan. Meanwhile, the president is moving to consolidate his gains.

"We've got more work to do. But what we're not going to do is unravel what has now been woven into the fabric of America," Obama said after the Supreme Court decision. "And my greatest hope is that rather than keep re-fighting battles that have been settled again and again and again, I can work with Republicans and Democrats to move forward."

One measure of the steepness of the hill Republicans have to climb: the sense inside the White House that the Supreme Court decision spelled the end of the threat to the law. After the Supreme Court decision came down, HHS Secretary Sylvia Mathews Burwell told reporters, "It was such a moment of relief."