Appearing on NBC's "Meet the Press," Romney said the president's "fundamental dishonesty ... has undermined the foundation of his second term."
"I think it is rotting it away," the former Massachusetts governor said.
"Had the president been truthful and told the American people that millions would lose their insurance and millions more would see their premiums skyrocket ... there would have been such a hue and cry against it, it would not have passed," Romney said.
Romney said Obama knew people would lose their insurance, but said the opposite during his campaign.
"As a centerpiece of his campaign and as a centerpiece of Obamacare ... saying time and time again that fundamental to his plan was the right people would have to keep their insurance plan, and he knew that was not the case," Romney said.
Software problems have impeded enrollment under the Affordable Care Act, making it "a miserably frustrating experience for way too many Americans," Health and Human Services Secretary Kathleen Sebelius told a House committee last week.
The criticism became more shrill when reports emerged that insurers were canceling coverage for many individual plan subscribers because the plans did not comply with federal law.
Some Democrats said Republicans were isolating information and using it out of context.
"If you have the kind of health care that disappears when you need it most, the Affordable Care Act says that has to end," said Massachusetts Gov. Deval Patrick, a Democrat.