WASHINGTON — The Supreme Court, over three dissents, rejected Arizona’s appeal Monday of a law that would have denied bail to immigrants here illegally who were arrested for a serious felony.

The measure had been adopted in 2006 by the state’s voters, and it said judges may not release on bail persons who have “entered or remained in the United States illegally” and were arrested for “serious felony offenses.”

Last year, however, the 9th Circuit Court of Appeals struck down the law and said that the Constitution’s protection for liberty applies to all persons in the United States and that people under arrest have a right to an individual hearing on whether they may be released before a trial.

Lawyers for Maricopa County asked the justices to reverse that decision, arguing that immigrants who were in the country illegally were not likely to show up for a trial if they were set free.

But after considering the case for several weeks, the court said it would not hear the appeal.

Justice Clarence Thomas, in dissent, said the court’s action “shows insufficient respect to the State of Arizona, its voters and its Constitution. And it suggests to the lower courts that they have a free rein to strike down state laws on the basis of dubious constitutional analysis.”