Democrats say that's just more wishful thinking, if not obsession.
Although Obama's law remains divisive, only 29 percent of the public favors its complete repeal, according to a recent Gallup poll. The business-oriented wing of the Republican party wants to move on to other issues. Americans may be growing weary of the health care fight.
"This is the law of the land at this point," said Michael Weaver, a self-employed photographer from rural southern Illinois who's been uninsured for about a year. "We need to stop the arguing and move forward to make it work."
It took him about a week and half, but Weaver kept going back to the healthcare.gov website until he was able to open an account and apply for a tax credit that will reduce his premiums. He's not completely finished because he hasn't selected an insurance plan, but he's been able to browse options.
It beats providing page after page of personal health information to insurance companies, Weaver said.
Under the new law, insurers have to accept people with health problems. Weaver is in his mid-50s, with high blood pressure and high cholesterol, but otherwise in good health. He says those common conditions made it hard for him to get coverage before.
Although Weaver seems to have gotten past the major website obstacles, he's still finding shortcomings. There's no place to type in his medications and find out what plans cover them. "I wish there was more detail, so you could really figure it out," he said.
Such a nuanced critique appears to be lost on congressional Republicans.
"(hash)TrainWreck: Skyrocketing Prices, Blank Screens, & Error Messages," screamed the headline on a press release Friday from House Speaker John Boehner, R-Ohio. A House hearing on the "botched Obamacare rollout" is scheduled for this coming week. GOP lawmakers want Health and Human Services Secretary Kathleen Sebelius to resign.
Administration officials, in their most detailed accounting yet of the early rollout, said Saturday that about 476,000 health insurance applications have been filed through federal and state exchanges. But the officials continue to refuse say how many people have enrolled in the insurance markets.
Without enrollment figures, it's unclear whether the program is on track to reach the 7 million people projecting by the Congressional Budget Office to gain coverage during the six-month sign-up period.
The president was expected to address the problems on Monday during a health care event at the White House. The administration has yet to fully explain what has gone wrong with the online signup system.
"To our Democratic friends: You own 'Obamacare' and it's going to be the political gift that keeps on giving," said Sen. Lindsey Graham, R-S.C.
"Irresponsible obsession," scoffs Rep. Sander Levin of Michigan, ranking Democrat on the Ways and Means Committee, which oversees much of the health law.
Democratic pollster Celinda Lake says she doesn't see how going after the health law rollout will help Republicans by the time of next year's election.
"Americans are technology optimists," said Lake. "You tell them the website has problems today, and they'll assume it will be better tomorrow. I mean, we're Americans. We can fix a website."
There may be a method to the GOP's single-mindedness.
Republicans are intent on making the health law an uncomfortable anchor around the neck of four Democratic senators seeking re-election in GOP-leaning states, weighing them down as they try to unseat them. Republicans need to gain six seats to seize the majority in the Senate, and any formula for control includes flipping the four seats.
Sens. Mark Pryor of Arkansas, Mary Landrieu of Louisiana, Mark Begich of Alaska and Kay Hagan of North Carolina will be facing voters for the first time since they were among the 60 Democrats who voted for the health law in 2009.
More than a year before the election, Republican Rep. Tom Cotton is airing an ad that criticizes Pryor for his vote, telling Arkansans that Pryor "cast the deciding vote to make you live under Obamacare." The commercial's final image shows Pryor with Obama, who took a drubbing in Arkansas last year.
"The bottom line is these candidates will have to answer for why they voted for this bill," said Rob Engstrom, senior vice president and national political director for the U.S. Chamber of Commerce.
If the website gets fixed, other problems may emerge. Republicans can still try linking 'Obamacare' to rising premiums, anemic job growth and broader economic worries.
Will the strategy work?
The chamber spent millions on ads in 2012 criticizing Senate incumbents such as Jon Tester of Montana and Bill Nelson of Florida for their health care votes, yet many of those candidates overcame the criticism and won re-election.
The economy, not health care, remains the top concern of voters. By putting opposition to the health care law ahead of all other priorities, economist Douglas Holtz-Eakin says tea-party conservatives may have overdone it.
"Obamacare was an effective campaign weapon," said Holtz-Eakin, and adviser to Republicans. "The question is, have they damaged it beyond its political viability?"