Pompeo told a news conference sanctions could extend to family members of ICC officials to prevent them from visiting the United States.
"We cannot, we will not stand by as our people are threatened by a kangaroo court,” Pompeo said.
ICC prosecutor Fatou Bensouda wants to investigate possible crimes committed between 2003 and 2014 including alleged mass killings of civilians by the Taliban, as well as the alleged torture of prisoners by Afghan authorities and, to a lesser extent, by U.S. forces and the CIA.
Trump has assailed the ICC, which was established in 2002 by the international community to prosecute war crimes, genocide and crimes against humanity. It has jurisdiction only if a member state is unable or unwilling to prosecute atrocities itself.