States that voted in Republican governors include Pennsylvania, Ohio, Michigan, Wisconsin, Kansas, Oklahoma, Wyoming, Tennessee, New Mexico, Iowa and Maine. Democrats reclaimed at least two seats from Republicans, in California and Hawaii.
House Republicans have pledged to repeal the law, which is designed to expand insurance to 32 million additional Americans, or at least choke off funding to implement it. Democrats in the Senate can block any repeal, and the defunding strategy faces roadblocks.
While governors can't avoid much of the law, they can throw sand in its gears and keep states out of involvement in a central part of it—new exchanges for selling insurance policies.
Wisconsin's Republican governor-elect, Scott Walker, met with lawmakers Wednesday to discuss how to minimize the state's participation in the law's expansion of Medicaid, the federal-state insurance program for the poor. He also wants to lean on private entities to run the insurance exchanges, where lower earners who qualify for tax credits and small businesses will shop for insurance starting in 2014.
Under Gov. Jim Doyle, a Democrat, Wisconsin ambitiously courted early health-law money, including funding for free birth control.
Mr. Walker is worried that the Medicaid expansion, initially paid for by the federal government, will be too costly once states must begin paying for a portion of it in 2017.
"Free money is not free," he said in an interview. "If we can't afford it, it doesn't matter how much of it is free."
Mr. Walker, along with new GOP governors in Wyoming and Oklahoma, said they planned to join in the legal fights against the law's requirement that most Americans carry insurance or pay a fine.
Plaintiffs in the largest suit, a 20-state effort led by Florida's Republican attorney general, plan to reach out to as many as six states with newly elected
Republicans to join the effort, according to a person familiar with the case, though it may be too late to join.
Health-care companies say GOP gains at the state level offer new opportunities to influence how the law is implemented.
"We have more Republican governors who may have interest in different models," Aetna Inc.'s Chief Executive Ronald Williams told analysts in a conference call Wednesday. So the discussion "may very well tip it more toward market-based solutions as opposed to others."
In a recent NBC News/Wall Street Journal survey, 51% of respondents said it would be acceptable to repeal the law. President Barack Obama said Wednesday he wanted to know which parts of the law Republicans don't like.
"When it comes to pre-existing conditions, is this something you're for or you're against?" he said at a news conference, referring to the mandate on insurers to cover those who already have ailments.
"Helping seniors get their prescription drugs—does that make sense or not?" he said.
Mary Fallin, a Republican elected Oklahoma governor, said she was emboldened by a largely symbolic 65%-35% vote in her state for a measure opposing the individual insurance requirement.
"I think the people of Oklahoma have spoken that they're concerned and they do not support the program of taking over the health-care system," Ms. Fallin said.
She said she was looking at whether Oklahoma can lean on an existing public-private program that provides insurance to the poor as an alternative to the law's Medicaid expansion.
"We certainly will try to minimize the impact the new federal law will have on the state of Oklahoma and certainly on our budget with unfunded mandates," Ms. Fallin said.
Should states opt out of the law's Medicaid expansion, they would have to remove themselves entirely from the Medicaid program—an unlikely outcome.
But Ray Scheppach, executive director of the National Governors Association, said states were likely to press the federal government to delay the expansion beyond 2014 given states' bleak budget picture.
Matt Mead, Wisconsin's Republican governor-elect, can appoint his own attorney general and insurance commissioner-two positions with ample influence over the law's enactment.Mr. Mead said in an interview that the current governor, a Democrat, has already begun preparing the state to operate the insurance exchanges, but he hasn't decided whether he will go forward with running them. The federal health law provides for state-based exchanges but allows states to leave operation to the federal government.