Conservative Republicans who forced the 16-day government shutdown by linking a funding measure to demands to dismantle or defund Obamacare now want to delay the creation of market exchanges -- a key component of the reforms.
The Obamacare exchanges create large pools of buyers to make coverage more affordable for the almost 50 million Americans without insurance.
Robust enrollment is vital to the program's success. The exchange effort also depends heavily on attracting younger, healthier participants, who would tax the health care system less but still pay premiums. That would help insurers offer lower rates to older Americans with more health care needs.
If younger people don't sign up, rates won't increase immediately but It could cause insurers to raise them in the future or drop out of the exchanges because they cannot make the economics work.
Carney said GOP calls to delay the individual mandate that requires people to obtain health coverage lacked "a sincere desire to improve the system" and instead were intended to undermine Obamacare.
However, some congressional Democrats -- especially those facing tough election battles next year -- are getting restless about the Obamacare problems.
Democratic Sens. Jeanne Shaheen of New Hampshire, Mark Begich of Alaska and Mark Pryor of Arkansas called for extending the enrollment period, while Rep. John Barrow of Georgia wants to delay the individual mandate.
The Obamacare exchanges are expected to include about 15% of Americans, with the rest of the insured population covered by employer or other group plans.
Republicans, including House Budget Chairman Paul Ryan of Wisconsin, also call for Sebelius to lose her job over the rollout problems.
Some Democrats also call for accountability, but they say the situation should first get fixed before assessing blame and firing anyone responsible.
Sebelius said she works "at the pleasure of the President" and is committed to her job.
"I think my job is to get this fully implemented and to get the website working right," she told Gupta.