The GOP's conservative base strongly opposes the individual mandate in the law, which requires people to obtain health insurance or pay a fine.
In its ruling, the Supreme Court said the mandate is constitutional under the government's taxing authority, and Republicans have jumped on that to characterize the provision as a tax increase on middle-class Americans.
Obama and Democrats respond that only people who can afford health insurance but choose not to get it would have to pay, amounting to about 1% of the population.
The health care issue has been among the most divisive of Obama's presidency. Conservative anger over the measure helped launch the tea party movement, and conservative groups joined with industry groups to fund a giant public pressure campaign against the legislation, which Democrats pushed through Congress with no Republican support.
Some provisions already have brought popular benefits, such as preventing insurance companies from denying coverage for children with pre-existing conditions.
The bulk of the health care law will take effect in 2014, including health insurance exchanges to provide options for individuals and small businesses to purchase coverage. Full implementation also will expand benefits, such as preventing insurance companies from denying coverage for adults with pre-existing conditions.
Romney, while pledging to repeal the health care law if elected, also says some provisions of it should be maintained, including the ban on denying coverage for pre-existing conditions.