WASHINGTON—President Barack Obama's health care overhaul has divided the nation, and Republicans believe their call for repeal will help them win elections in November. But the picture's not that clear cut.

A new AP poll finds that Americans who think the law should have done more outnumber those who think the government should stay out of health care by 2-to-1.

"I was disappointed that it didn't provide universal coverage," said Bronwyn Bleakley, 35, a biology professor from Easton, Mass.

More than 30 million people would gain coverage in 2019 when the law is fully phased in, but another 20 million or so would remain uninsured. Bleakley, who was uninsured early in her career, views the overhaul as a work in progress.

The poll found that about four in 10 adults think the new law did not go far enough to change the health care system, regardless of whether they support the law, oppose it or remain neutral. On the other side, about one in five say they oppose the law because they think the federal government should not be involved in health care at all.

The AP poll was conducted by Stanford University with the Robert Wood Johnson Foundation. Overall, 30 percent favored the legislation, while 40 percent opposed it, and another 30 percent remained neutral.

Those numbers are no endorsement for Obama's plan, but the survey also found a deep-seated desire for change that could pose a problem for Republicans. Only 25 percent in the poll said minimal tinkering would suffice for the health care system.

Brian Braley, 49, a tech industry worker from Mesa, Ariz., wants Washington to keep its hands off. "I think it's a Trojan horse," Braley said of the health care law. "It's a communist, socialist scheme. All the other countries that have tried this, they're billions in debt, and they admit this doesn't work."

It may well satisfy people who share Braley's outlook if Republicans succeed in tearing out what they dismiss as "Obamacare" by the roots. But GOP leaders would still find themselves in a quandary.

Republicans "are going to have to contend with the 75 percent who want substantial changes in the system," said Stanford political science professor Jon Krosnick, who directed the university's participation.

"Republican legislators' passion to repeal the legislation is understandable if they are paying attention to members of their own party," Krosnick added. "But if they want to be responsive to all Americans, there are more Democrats and independents than there are Republicans."