The United States has always regarded the court warily, fearing that it would be used against American troops as a way to subvert Washington’s foreign policy decisions. President Bill Clinton signed the 1998 Rome treaty establishing the court, despite noting its “significant flaws.”

Congress never ratified the treaty, but former Presidents George W. Bush and Barack Obama at times gave the court at least tacit support. The Bush administration endorsed its investigation into atrocities in Sudan. The Obama administration provided intelligence and Mr. Obama personally encouraged other countries to cooperate with it.

While two-thirds of the world’s nations are members of the court, that coalition has been tested recently. Burundi withdrew last year — although Burundian officials still face possible prosecution for crimes committed while it was a member. And President Rodrigo Duterte of the Philippines announced plans this year to quit the court amid an investigation into allegations that officials committed mass murder and crimes against humanity during a narcotics crackdown.

Several African nations have at times threatened to quit. And President Vladimir V. Putin of Russia, which was never a member of the court, withdrew his signature from the treaty when the court announced it would investigate Moscow’s military involvement in Ukraine and possible war crimes in Georgia.

“The court is vulnerable,” said Carsten Stahn, an international law professor at Leiden University here. “Bolton is using that vulnerability to attack.”

Mr. Bolton has fiercely opposed the court since its inception and his remarks represented a warning to the court over its apparent intention to investigate war crimes in Afghanistan, including the torture and abuse of prisoners by C.I.A. officers. Though Congress has documented many of those abuses, nobody has been prosecuted.