WASHINGTON — The Supreme Court on Monday delivered a split decision on Arizona’s tough 2010 immigration law, upholding its most hotly debated provision but blocking others on the grounds that they interfered with the federal government’s role in setting immigration policy.

The court unanimously sustained the law’s centerpiece, the one critics have called its “show me your papers” provision, though they left the door open to further challenges. The provision requires state law enforcement officials to determine the immigration status of anyone they stop or arrest if they have reason to suspect that the individual might be in the country illegally.

The justices parted ways on three other provisions, with the majority rejecting measures that would have subjected illegal immigrants to criminal penalties for activities like seeking work.

The ruling is likely to set the ground rules for the immigration debate, with supporters of the Arizona law pushing for “show me your papers” provisions in more states and opponents trying to overturn criminal sanctions for illegal immigrants.

Writing for the majority, Justice Anthony M. Kennedy said, “Arizona may have understandable frustrations with the problems caused by illegal immigration while that process continues, but the state may not pursue policies that undermine federal law.”

Justice Antonin Scalia summarized his dissent from the bench, a rare move that indicated his deep disagreement. Rarer still, he criticized a policy that was not before the court: President Obama’s recent announcement that his administration would not deport many illegal immigrants who came to the United States as children.

Justice Scalia’s point was a narrow one — that the states should have the right to make immigration policy if the federal government is not enforcing its own policies — but it continued a charged back and forth between the conservative justices and Mr. Obama. In his 2010 State of the Union address, Mr. Obama criticized the court’s Citizens United campaign finance ruling, which the court reiterated in a separate ruling on Monday.

The court also announced that it was extending its term until Thursday, signaling that it would issue its much-anticipated ruling on Mr. Obama’s health care law then.

Both Mr. Obama and Mitt Romney, the presumptive Republican presidential nominee, quickly responded to the immigration ruling. Mr. Romney — traveling, by coincidence, in Arizona — said in a brief statement that states had the right and the duty to secure their borders.

Mr. Obama emphasized his concern that the remaining provision could lead to racial profiling, an issue that the court may yet consider in a future case. “No American should ever live under a cloud of suspicion just because of what they look like,” Mr. Obama said in a statement, adding that he was “pleased” about the parts that were struck down.

In her own statement, Gov. Jan Brewer of Arizona, a Republican, said she welcomed the decision to uphold what she called the heart of the law. The decision, she said, was a “victory for the rule of law” and for “the inherent right and responsibility of states to defend their citizens.”

Still, the ruling was a partial rebuke to state officials who had argued that they were entitled to supplement federal efforts to address illegal immigration.

The Obama administration argued that federal immigration law trumped — or pre-empted, in legal jargon — the state’s efforts. Last year, the United States Court of Appeals for the Ninth Circuit, in San Francisco, blocked the four provisions on those grounds, including the one the Supreme Court upheld.

In its challenge, the administration did not argue that it violated equal-protection principles. At the Supreme Court argument in April, Solicitor General Donald B. Verrilli Jr. acknowledged that the federal case was not based on racial or ethnic profiling.

In the majority opinion, Justice Kennedy wrote that the ruling did not foreclose other “constitutional challenges to the law as interpreted and applied after it goes into effect.”

Meanwhile, Attorney General Eric H. Holder Jr. said on Monday that the federal government would “continue to vigorously enforce federal prohibitions against racial and ethnic discrimination.”

Five other states have enacted tough measures to stem illegal immigration, more or less patterned after the Arizona law: Alabama, Georgia, Indiana, South Carolina and Utah. But most states avoided creating new crimes for immigration violations, as Arizona did in two provisions that were struck down.

Lower courts have stayed the carrying out of parts of those laws, and they will now revisit those decisions.

In upholding the requirement that the police ask to see people’s papers, the court emphasized that state law enforcement officials already possessed the discretion to ask about immigration status. The Arizona law merely makes that inquiry mandatory if the police have reason to suspect a person is an illegal immigrant.

In a concurring opinion, Justice Samuel A. Alito Jr. called the administration’s attack on the provision “quite remarkable.” Read more from the New York Times