ASHINGTON, March 21 - A domestic-violence nightmare provided the basis for an intense and sober Supreme Court argument on Monday, as the justices considered whether the police may be held liable for damages for failing to enforce a court-issued order of protection.

A federal appeals court has permitted a $30 million lawsuit against the town of Castle Rock, Colo., brought by a woman whose former husband kidnapped their three young daughters and murdered them while she fruitlessly sought the help of the local police. The woman, Jessica Gonzales, had obtained an order of protection limiting her ex-husband's contact with the children to specified times.

In the appeals court's view, a provision of Colorado law telling law enforcement officials that they "shall use every reasonable means to enforce" such orders entitles individual citizens to a response from the police when an order is violated. The law has the effect of creating a "property interest" in police protection, which cannot be denied without "due process," the United States Court of Appeals for the 10th Circuit, in Denver, ruled last year.

On the basis of the justices' responses during Monday's argument, the Supreme Court appeared unlikely to agree. "This is such a new sort of requirement you're seeking to have us develop here," Justice Sandra Day O'Connor said to Brian J. Reichel, the lawyer for Ms. Gonzales.

"It would be a major step, wouldn't it?" Justice O'Connor asked, adding that she knew of "no case" that supported the appeals court's approach.

In a 1989 decision, DeShaney v. Winnebago County Social Services Department, the Supreme Court ruled that generally, the government does not have a constitutional obligation to protect a person from harm at the hands of a private citizen. In that case, the court found that a county social services department in Wisconsin did not breach any constitutional duty to a young boy when it returned him to the custody of an abusive father, who then beat him, leaving him brain-damaged.

The appeals court last year, and Ms. Gonzales's lawyer on Monday, tried to skirt the boundaries of that precedent by framing the issue not as a substantive right to protection, but as a right not to be deprived of protection without procedural guarantees - "procedural due process," in legal jargon.

Ms. Gonzales was entitled to an "objective, thoughtful, reasoned evaluation of her complaint of nonenforcement," Mr. Reichel asserted. He said the Castle Rock police owed her an immediate explanation of why they refused her repeated requests for help after the kidnapping, on June 22, 1999. Nearly 12 hours after he took the children, her former husband, Simon Gonzales, arrived at the police station, opened fire and was shot dead at the scene; in the back seat of his truck were the bodies of the three girls.

The case, Town of Castle Rock v. Gonzales, No. 04-278, has become a rallying point for individuals and organizations interested in protecting victims of domestic violence. They and some law enforcement organizations argue that protection orders mean little unless the police have procedures for handling violations.

On the other side, the Bush administration joined Castle Rock's appeal in arguing that under a long-accepted "background principle," individuals cannot go to court to invoke a right to law enforcement.

In court on Monday, Justice John Paul Stevens asked John P. Elwood, an assistant solicitor general who argued in support of the Colorado town, whether the police "have any duty at all" to enforce an order of protection.

Mr. Elwood replied, "I don't believe the police have an actionable duty."

Justice Stevens then asked whether the police could adopt a policy that said, "Ignore all orders of this kind."

In that instance as well, Mr. Elwood said, there would be no individual right to challenge such a policy.

Justice Anthony M. Kennedy asked whether a police officer who came upon four people beating someone could stand by without intervening because "he enjoys watching a fight."

Under the DeShaney precedent, Mr. Elwood replied, there is no constitutional obligation for the police to act unless it was the government that had placed the victim in harm's way.

Castle Rock's lawyer, John C. Eastman, told the justices that while "what happened here is undeniably tragic," Ms. Gonzales received "whatever process might be due." Mr. Eastman said the language in Colorado's law, informing the police that they "shall use every reasonable means to enforce" a protective order, did not deprive the police of discretion to decide what action might be "reasonable."

This led Justice Ruth Bader Ginsburg to ask, "Does discretion include discretion to do nothing?"

Despite the justices' evident sympathies for Ms. Gonzales, they appeared reluctant to accept her argument.

"It's outrageous what happened, a terrible tragedy," Justice Stephen G. Breyer told her lawyer, Mr. Reichel. But Justice Breyer said that if the police were required to give a reason for not acting, the reason could simply be that "we're too busy," and that courts would not second-guess that determination.

"What you need is instructions to the police that when a child is missing, you don't wait," Justice Breyer continued. "I don't see how the route you take gets at that problem."

Mr. Reichel replied that the constitutional problem was not the delay but the failure by the police to explain their nonaction.

"What good would that have done her?" Justice David H. Souter asked. "What is the social value?"

"Your argument could apply to every statute," Justice Souter said, and would create "a completely nonadministratable situation."