The health care overhaul spearheaded by President Obama has allowed 6.6 million young Americans who would otherwise not have insurance to get coverage through their parents, according to a new report by the Commonwealth Fund.

One of the law's most popular measures allows young adults to stay on their parents' insurance plans until their 26th birthdays. Even with that safeguard, about two in five Americans between the ages of 19 and 29 went without insurance for all or part of 2011.

Despite the law's success in covering young adults -- typically a demographic that is among the least likely to obtain insurance -- a majority of Americans still favor doing away with some or all of the law. A CBS/New York Times poll found that 41 percent of Americans support the Supreme Court overturning the law completely (oral arguments on the law's constitutionality happened in March, and the court is expected to render a decision this month).

The poll also underscored the fact that sections of the law remain popular, such as the young adult coverage piece and a section barring insurance companies from denying coverage to people with pre-existing conditions. Twenty-seven percent of respondents supported eliminating the mandate that all Americans obtain health insurance but keeping the rest of the law.

Only 24 percent of Americans said they favor keeping the law in its entirety. The individual mandate has come to solidify the widespread perception that the law impinges on personal freedom, a central part of the Republican party's critique. Likely Republican presidential nominee Mitt Romney has echoed that criticism despite pushing for an individual mandate when he was governor of Massachusetts.