Debate over the law is expected to dominate the midterm elections. Attacks on the law are featuring prominently in campaigns across the country, and Republican lawmakers have continued to push for the law’s replacement. Mr. Obama cites the millions who have signed up for insurance as evidence of the law’s success.

Opposition to the president runs deep in the four states. In Arkansas, for example, 64 percent of respondents said they disapproved of his handling of health care. Still, the poll revealed that 60 percent of people in North Carolina and 52 percent of people in Louisiana and Kentucky would rather the law be made better than taken away. In Arkansas, 48 percent were in favor of improvements and 46 percent for repealing the law and replacing it.

National polls have identified the sentiment before, but the fact that it surfaced in four states that voted Republican in the last election underscored just how deeply it is held.

Democrats in Congress have requested changes to the law, arguing that earlier major pieces of social legislation, like Medicare, underwent revisions. But Republicans, who control the House, have refused to allow fixes, pushing for repeal and replacement.

Views fell along predictably partisan lines, with large shares of Republicans in all four states preferring repeal and replacement and most Democrats opting for improvements. Independents were much more evenly divided, and in North Carolina and Louisiana, majorities preferred improvements.

Perhaps more surprising were views on Medicaid, the medical program for the poor that only about half the states expanded under the law. When asked what states should do about Medicaid, the respondents offered substantial support for expansion.

In Arkansas, a majority backed it, while less than a quarter opted for what the state actually chose — a “private option” under which the state used federal money to purchase private insurance for the poor through the new health care marketplace. Even among Republicans in the state, half supported Medicaid expansion while 21 percent liked the private option. In Louisiana and North Carolina, more than half of people agreed that their state should expand Medicaid and use federal funding to cover most of the cost.