Sen. Lindsey Graham, R-S.C., said on ABC's "This Week" that a compromise that would raise the debt ceiling and restore full government operations did not translate to a capitulation on healthcare reform.
"We are in a free-fall as Republicans, but Democrats are not far behind," Graham said.
"It is unrealistic to expect us to defund or delay Obamacare by shutting the government down. But the fight on Obamacare is far from over. After this mess is behind us, Obamacare will be a liability for Democrats."
Graham cited polls that showed a high degree of voter dissatisfaction with both parties, and contended President Obama and Senate Democrats were now the ones who refused to compromise in order to end the shutdown and raise the debt ceiling.
"The sooner this is over, the better for us [Republicans]," Graham said. "And to our Democratic friends, you own Obamacare and it's going to be the political gift that keeps on giving. The shutdown will be old news next year; Obamacare's faults will be front and center in 2014 if we don't screw this up."
Rep. Keith Ellison, D-Minn., said there were areas of Obamacare that could be fine-tuned, but that it was unrealistic to expect the administration to negotiate with the shutdown and possible debt default hanging over its head.
"It will be us giving and taking and them giving and taking, not a situation like we have now, which is we will stop inflicting pain on America if you get rid of healthcare, which brings insurance reform and access to millions of Americans," Ellison told ABC. "We can't do that."