“This deals with hundreds of thousands of Missourians, it deals with their health care, it deals with billions of dollars, and we will be involved in a process that defines the best fit for our state and respects the sovereignty of our state and the individuality of our state,” said Gov. Jay Nixon of Missouri, a Democrat who is facing re-election this year, as well as a Republican-led legislature.

Gov. Bob McDonnell of Virginia, the chairman of the Republican Governors Association, has been one of the sharpest critics of the health care law, but he did not rule out the possibility that he might ultimately decide to expand his state’s Medicaid program. “Honestly, I don’t think it’s responsible for my state, fully, to make the decision now,” he said.

Many Republican governors who opposed the law delayed taking action on it before the court ruled. Several, including Gov. Scott Walker of Wisconsin, now say they will wait to act until after November in the hopes that Republicans will win the presidency and the Senate and dismantle it. But Mr. Walker did not wholly reject the idea of expanding Medicaid. “We’re not ruling it one way or the other,” he said on Saturday after a meeting about controlling health care costs.

Once governors make up their minds, many could face battles with their legislatures. Gov. Mark Dayton of Minnesota, a Democrat, has made expanding Medicaid a top priority, but he faces some strong resistance from the state’s Republican-controlled Legislature. And Gov. Chris Christie of New Jersey, a Republican who has expressed doubts about expanding the program, faces a Legislature controlled by Democrats, many of whom want to implement the new law.