White House press secretary Jay Carney on Monday said Obamacare is “not failing” in the face of criticism that the administration botched the rollout of the new website responsible for enrolling consumers in the insurance exchanges.

“This system is not failing. Hundreds of thousands of Americans are submitting their applications successfully to get into the system and enroll in Obamacare, and they are doing it through a variety of means, through state exchanges and through the call-in centers and that's going to continue,” said Carney in an interview on CNN's “Piers Morgan Live.”

Critics of Obamacare have seized on the technical issues, arguing that the administration was unprepared for the rollout of Obama’s signature domestic achievement. Many Republican lawmakers have pressed for delaying other aspects of the law -- in particular, the individual mandate which would fine consumers who do not purchase health insurance.

The White House on Monday made a push to reassure the public that it would fix glitches in the new Obamacare websites launched Oct. 1. The administration has said they are implementing a "tech surge" to fix the issues and have blamed a high volume of interest in the websites for the glitches.

President Obama on Monday spoke at the White House and said “nobody’s madder” than him about the missteps, but called the healthcare law a “good deal” and said it was already benefiting Americans.

Some GOP lawmakers have also called for Obama to fire Health and Human Services Secretary Kathleen Sebelius. HHS said on Monday that Sebelius would testify before Congress as early as “next week.”

Carney, though, said the focus should be on making the insurance websites work.

“It shouldn't be about making heads roll or firing people,” he said.

Carney said Obama was focused on making sure those responsible had the website up and running quickly.

“The accountability he is looking for is the accountability of making sure that everybody who has expertise in this matter is focused on fixing it, not focused on making heads roll. It’s not the time right now to focus on that,” he added. “The time is to get these problems fixed and make sure that the system is working most efficiently for the American people.

“That's what his focus is now, not pointing fingers and making blame,” said Carney.