But John Kavanagh, a Republican state legislator, said the federal government had not done enough. “Arizona has a serious problem with illegal immigration, being one of the leading illegal entry states, so protecting the credibility of our election system requires that we exclude illegal aliens and any other noncitizen from voting,” he said. “Not being able to request proof makes that impossible.”

The decision, with its lopsided vote, is not an indication that the court will always be sympathetic to claims of voter suppression. Its decision on the constitutionality of the Voting Rights Act of 1965 is expected by the end of the month, and it may limit what its supporters say is an important tool in protecting minority voters.

Nor has the Supreme Court been uniformly hostile to Arizona’s efforts to address what lawmakers there say is a crisis in illegal immigration. Last year, it upheld one part of a tough 2010 state immigration law even as it endorsed broad federal power over immigration. In 2011, it sustained a different law that imposed harsh penalties on businesses that hired illegal workers.

On Monday, Justice Antonin Scalia, writing for the majority in Arizona v. Inter Tribal Council of Arizona, No. 12-71, said a federal law requiring states to “accept and use” a form displaced an Arizona law requiring various kinds of proof of citizenship.

The law, the National Voter Registration Act of 1993, allows voters to register using a federal form that asks, “Are you a citizen of the United States?” Prospective voters must check a box for yes or no, and they must sign the form, swearing under the penalty of perjury that they are citizens.