Starting in 2014, the law requires most Americans to have health insurance. It also offers subsidies to help people pay for insurance bought through markets known as insurance exchanges. At issue is whether the subsidies will be available in exchanges set up and run by the federal government in states that fail or refuse to establish their own exchanges. Critics say the law allows subsidies only for people who obtain coverage through state-run exchanges. The White House says the law can be read to allow subsidies for people who get coverage in federal exchanges as well. The law says that “each state shall” establish an exchange. But Washington could be running the exchanges in one-third to half of states, where local officials have been moving slowly or openly resisting the idea. The dispute has huge practical implications. The Congressional Budget Office predicts that 23 million uninsured people will gain coverage through exchanges and that all but five million of them will qualify for subsidies, averaging more than $6,000 a year per person. Subsidies, in the form of tax credits, will be available to people with incomes from the poverty level up to four times that amount ($23,050 to $92,200 for a family of four). Some supporters of the law say Congress may have made a mistake in drafting this section. But, they add, the intent of Congress is clear: subsidies should be available in federal as well as state exchanges. The law says that subsidies will be provided to residents of a state to help defray the cost of health plans offered “through an exchange established by the state.” A rule issued by the Obama administration allows tax credits for insurance bought in either a state or a federal exchange. Political brawling over health care will continue this week on Capitol Hill. The House plans to vote on a Republican measure to repeal the 2010 health care law, Mr. Obama’s most significant legislative achievement. Democrats, who control the Senate, say repeal efforts have no chance of success there. Representative Phil Roe, Republican of Tennessee, said the rule on premium subsidies “contradicts the explicit statutory language” of the Patient Protection and Affordable Care Act. Mr. Roe and another Tennessee Republican, Scott DesJarlais, have introduced a bill to nullify the rule, issued by the Internal Revenue Service. Douglas H. Shulman, the I.R.S. commissioner, defended the rule as consistent with the intent of Congress. “The statute,” he said, “includes language that indicates that individuals are eligible for tax credits whether they are enrolled through a state-based exchange or a federally facilitated exchange.” However, Senator Orrin G. Hatch of Utah, the senior Republican on the Senate Finance Committee, said the Obama administration was usurping the role of Congress and rewriting the law to provide tax credits through federal exchanges. White House officials have repeatedly underestimated opposition to the health care law. They predicted that public support for the law would grow as people learned more about it. They minimized lawsuits challenging the constitutionality of the requirement for most Americans to carry health insurance. They predicted that states would embrace the opportunity to insure more of their residents by expanding Medicaid and creating insurance exchanges, but a number of states have balked. James F. Blumstein, a professor of constitutional and health law at Vanderbilt University, said the dispute over subsidies involved a serious legal issue. “The language of the statute is explicit,” Mr. Blumstein said. “Subsidies accrue to people who obtain coverage through state-run exchanges. The I.R.S. tries to get around that by providing subsidies for all insurance exchanges. That interpretation will almost certainly be challenged by someone.” The most likely challenger, Mr. Blumstein said, is an employer penalized because one or more of its employees receive subsidies through a federal exchange. Employers may be subject to financial penalties if they offer no coverage or inadequate coverage and at least one of their full-time employees receives subsidies. Michael F. Cannon, director of health policy studies at the libertarian Cato Institute, said the link between subsidies and penalties was a crucial part of the law. “Those tax credits trigger the penalties against employers,” Mr. Cannon said. If workers cannot receive subsidies in states with a federal exchange, their employers cannot be penalized, he said. That, in turn, would hobble federal efforts to get employers to offer coverage in those states, Mr. Cannon said. Prof. Timothy S. Jost, an expert on health law at Washington and Lee University, said Congress had made “a drafting error” that should be obvious to anyone who understands the new health care law. “There is no coherent policy reason why Congress would have refused premium tax credits to the citizens of states that end up with a federal exchange,” said Mr. Jost, who supports the law.