WASHINGTON >> In a case that could further roil relations with Mexico, the Supreme Court considered on Tuesday whether the parents of a 15-year-old Mexican boy killed by a U.S. border guard may sue in U.S. courts.

The teenager, Sergio Hernández Guereca, was on the Mexican side of the border when the guard, Jesus Mesa Jr., standing about 60 feet away in U.S. territory, shot him in the head.

A lawyer for the youth’s parents faced two kinds of hurdles at Tuesday’s argument. Even justices sympathetic to the parents appeared to be worried about the implications of a ruling in their favor, which would extend constitutional protections beyond the nation’s borders. And some justices suggested that any solution should come from Congress rather than the courts.

The lawyer, Robert C. Hilliard, said he sought only a narrow ruling. The Constitution applies abroad, he said, only when the challenged conduct starts in the United States, when the killer and the victim are both civilians, when the constitutional right at issue is the right to life, and when the foreign government supports the suit.

Chief Justice John G. Roberts Jr. said that this test as proposed by Hilliard was opportunistic rather than principled. “That’s a test that, surprisingly, fits the exact facts of your case,” he said with a note of sarcasm.

The chief justice said a ruling in the parents’ favor could have implications in many areas, including “a drone strike in Iraq where the plane is piloted from Nevada.”

Justice Stephen G. Breyer said the parents had “a very sympathetic case.” But he said it was hard to rule in their favor without considering the decision’s implications for many other cases.

Justice Samuel A. Alito Jr. pressed Hilliard along those lines. “You need to give us a principle that is workable,” Alito said.

Hilliard responded that there were special problems on the Mexican border.

“The Border Patrol is 44,000 strong along our southwest border,” he said. “We’ve had 10 shootings across the border.”

The case, Hernandez v. Mesa, No. 15-118, started in 2010, with four Mexican boys playing in the dry bed of the Rio Grande that separates El Paso from Juárez, Mexico. The international borderline, unmarked, runs through the middle of the culvert.

The boys dared one another to run up a concrete incline and touch the barbed wire of the U.S. border fence. Mesa, the border guard, grabbed one of them.

Sergio fled, and he made it back to Mexico before Mesa shot him.

A statement issued by the FBI the day after the shooting suggested that Mesa had been justified in using deadly force because Sergio had thrown rocks at him. But when the Justice Department closed its investigation into the shooting in 2012, it said that “a group of smugglers” had thrown rocks, but it did not accuse Sergio of doing so. Cellphone videos appear to show that he was trying to run and hide.

Mexican authorities charged Mesa with murder. But the United States has refused to extradite him. The Justice Department declined to prosecute Mesa, and it said he had not violated government policies concerning the use of force.

At the argument on Tuesday, Breyer suggested a way to rule for Sergio’s parents based on the court’s 2008 decision in Boumediene v. Bush, concerning people detained at the military prison in Guantánamo Bay, Cuba. That decision allowed detainees to invoke the Constitution based on the United States’ practical control over the naval base there.

The culvert where Sergio was shot could be said to be a similar kind of place, Breyer said. It “is a joint effort of Mexico and the United States,” he said. “So this is not just like a fence. It is an area of two fences, and between those two areas is joint exercise of border maintenance authority.”

Breyer added that both Mexican and American children played in the culvert.

“The dividing line isn’t even marked on the ground,” Justice Elena Kagan said. “You can’t tell on the ground where Mexico ends and the United States begins.”

Randolph J. Ortega, a lawyer for Mesa, said borders mattered. “Wars have been fought to establish borders,” he said. “The border is very real.”

But Justice Ruth Bader Ginsburg said it was odd that a Mexican boy killed on the U.S. side of the border would have rights while one killed just feet away across the border would not.

“That doesn’t make a whole lot of sense, does it, to distinguish those two victims?” she asked.

As the argument progressed, it seemed clear that the court’s four more liberal justices were inclined to vote to allow the parents to sue. But it was less certain that they could capture the crucial fifth vote of Justice Anthony M. Kennedy, who did not seem to think the question was for the courts to decide.

“You’ve indicated that there’s a problem all along the border,” Kennedy told Hilliard. “Why doesn’t that counsel us that this is one of the most sensitive areas of foreign affairs where the political branches should discuss with Mexico what the solution ought to be?”

Edwin S. Kneedler, a lawyer for the federal government, agreed. “You have a cross-border incident which necessarily gives rise to foreign relations problems, which are committed to the political branches,” he said.

© 2017 The New York Times Company