At the same time, President Obama was launching a final push to promote the law in advance of a major provision going into effect next Tuesday.
Speaking at a Maryland community college in suburban Washington, he decried the "misinformation" swirling around the law but said Republicans have been unsuccessful "at every step" in their efforts to stop it.
Government-regulated insurance "exchanges" are set to launch on Oct. 1. "The closer we got to this date, the more irresponsible folks opposed to this law have become," Obama claimed.
But House Republicans stood by their strategy on Thursday. "The American people don't want the president's health care bill," House Speaker John Boehner said.
This comes as the Senate deals with a measure that would keep the government open past Sept. 30 in exchange for defunding the health law. The Senate voted Wednesday to advance the bill and, if Democrats can clear one final hurdle on Friday, Majority Leader Harry Reid plans to strip out the ObamaCare measure and send it back to the House.
At that point, House Republicans would be faced with a choice.
They could continue to push for anti-ObamaCare provisions in the budget bill. This, though, risks a government shutdown on Oct. 1, as Democrats refuse to go along with any budget bill that defunds the health law. GOP leaders are also exploring adding face-saving options -- like the repeal of a tax on medical devices, which many Democrats also oppose -- to the stopgap spending bill.
But as an alternative path, Republicans could simply pass a "clean" budget bill, and immediately shift their attention to the fight over the debt ceiling -- using that as leverage to push for a one-year delay of the health law.
This is arguably a much bigger fight with much bigger implications.
Treasury officials say the U.S. government exhausts its borrowing capability on Oct. 17. Failure to raise the debt ceiling by then could risk a U.S. default.
Republicans, though, are hoping to use that doomsday scenario to convince Democrats to relent and delay the health law by a year.
They argue the administration has already delayed the requirement on mid-sized and large businesses to provide insurance to workers, and that the government should extend that to the rest of the law.
"I think we ought to really force the effort to delay this individual mandate for at least a year, just as the president unilaterally decided to delay the employer mandate," Sen. John Barrasso, R-Wyo., told Fox News, though he wants to achieve that through the budget bill.
"I think it's very important to give American people the same relief that the president unilaterally decided to give to others," he said.
But the Obama administration is already gearing up for a major launch of the so-called insurance exchanges on Oct. 1, and has no interest in delaying the law. The White House is equally insistent that Congress approve the increase in the debt ceiling.
"The president's not going to accept that proposition. He's just not. We will not negotiate over Congress' responsibility to pay the bills that Congress racked up," White House Press Secretary Jay Carney said Wednesday.
Aside from the ObamaCare provision, the GOP's demands on the debt limit involves far less dramatic spending cuts than Republicans demanded from Obama in a debt showdown two years ago. Then, Republicans extracted $2.1 trillion in cuts over a decade for a similar increase in the borrowing cap. Now, GOP leaders are mulling a 14-month borrowing increase that would increase the debt ceiling by almost $1 trillion but are considering only modest cuts, like an increase in the contribution federal workers make to their pensions.