Pence cited the "power of the purse" -- Congress' prerogative to appropriate funds to federal agencies -- as a key tool at the Republicans' disposal if they win back the House. That's not just bluster.
"The most serious, yet realistic, possibility is precisely the one that you're suggesting: what the Republicans can do through appropriations bills," says Paul van de Water, a health care expert at the Center on Budget and Policy Priorities.
In short, implementing the health care law costs money. "Some money was provided in the health reform bill itself, but not by any means all the administrative funding that will be needed," van de Water said. "If HHS and Treasury don't get appropriations they need to run the law well, that could be a real problem. It's not sexy but it's serious."
This can work one of few ways. House Republicans, in negotiations with the Senate, could demand appropriation levels beneath what's necessary to effectively implement the law. If the two chambers reach an agreement -- even an agreement that leaves the health care law cash strapped -- Obama would be hard pressed to issue a veto. "It's hard for the president to veto a bill because it doesn't provide enough money."
"In theory [they] could cut the funding 10 percent, 15 percent, 20 percent," says Congressional expert Norm Ornstein of the American Enterprise Institute. "The problem is, you could do a lot of damage in a lot of different places."
But things could shake out differently. An agreement might not be reached, for instance. Or, similarly, Republicans could simply "refuse to fund the entire Labor-HHS appropriations bill, or...pass an appropriation for Labor-HHS that does not include any funds for implementation of the health care plan," as Ornstein put it.
"They could really bollocks things up if they say 'none of the funds in this bill can be used to administrate the Affordable Care Act," echoes van de Water.
That could lead to a veto and then a showdown between the White House and the Hill, mimicking the 1995 standoff between Bill Clinton and then-House Speaker Newt Gingrich.
That's a double-edged sword. The 1995 shutdown blew back in the GOP's face politically -- but if the goal is to damage the health care law, it would be an effective tactic. Even if Republicans ultimately relent, weeks or months of delay would take its toll on the implementation of health care reform.
"If they try to go to that extreme, then I think the president's hand is much stronger," van de Water says. "The question is how far are they willing to go, and what the outcome would be."
There are other tricks the GOP could pull, too. "A second thing that they can do is hold a bunch of hearings and try to tie HHS and CMS into knots, by subpoenaing docs calling in of key figures to testify. In effect, deliberate sabotage to gum up the works," Ornstein adds.
It's all but impossible to get Democrats to discuss this threat openly -- it's election season, and they have to hew tightly to the line that a GOP takeover of the House is impossible. But it's not.
The health care law, and therefore Obama, will ultimately be judged by how successful it is. If Republicans retake the House, they're promising to jeopardize that legacy.