The administration is furiously trying to rectify another Obamacare issue: faulty enrollment data that could soon become a significant headache for consumers and insurers.

The latter say much of the information they have received is practically useless, which means some consumers might not be able to get access to benefits on Jan. 1, the date their coverage is scheduled to take effect.

Dr. Bill Grace, founder of Grace Oncology in NYC and an assistant professor of clinical medicine at New York Medical College, told CNBC Monday night that the enrollment problems also are causing significant problems for doctors.

"Right now that back end is in chaos. … We don't even know if we should sign up for these plans, because we have a suspicion that many of [them] offer very little reimbursement or offer a loss to the physicians who are going to see these patients," Grace told Larry Kudlow on CNBC's "Kudlow Report".

"No physician wants to take money out of his pocket and put it in the government's."

(Read more: Obamacare enrollment error rate still secret as more sign up)

The Centers for Medicaid & Medicare Services, which is responsible for running the federal Obamacare website, has refused to reveal how many sign-ups have software-related errors that could delay enrollees' coverage.

The administration released a 50-state report Tuesday morning saying that nearly 1.5 million people had been found eligible for Medicaid during October, far outpacing enrollment eligibility in subsidized private coverage. But as Obamacare puts more people in the Medicaid system, Grace said fewer doctors are treating Medicaid patients.

"The problem is, who's going to care for you when you have Medicaid?" he said. "For an hour-and-15-minute evaluation of a cancer patient, I get $6.50. That won't even pay for the electricity. What kind of physician will you get ... what kind of specialist will you get?"

Doctors are "already leaving in droves from accepting Medicare," he said. "They're going to concierge practices. They're limiting the number of Medicare patients they take. They're taking only commercial insurance plans, and the premier physicians are taking no insurance at all."

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Obamacare's website troubles have resulted in enrollment numbers significantly lower than administration officials had hoped. It's questionable whether the program will reach the 7 million sign-ups the Congressional Budget Office predicted. The sign-up period runs through March.

Grace told CNBC that the biggest problem with Obamacare is that patients don't have skin in the game. He offered this health-care fix as an alternative.

"I'd give everybody at birth a health savings account, because it makes everybody a capitalist. People make poor health care choices and want somebody else to bail them out. When they own the plan, they begin to make healthy ... choices. "

—By CNBC's Ben Thompson. Follow him on Twitter @BenThompson00.