"While victims' rights should be the very top priority of the United States government, throwing roadblocks in front of the ICC's investigation undermines justice not only for abuses committed in Afghanistan, but also for the millions of victims and survivors throughout the world who have experienced the most serious crimes under worldwide law".
The Hague-based court, the first global tribunal for war crimes, issued a statement saying it would continue to operate "undeterred" by the United States action.
Pompeo said "implementation" of the policy has already begun but he did not provide any details, citing confidentiality surrounding visa applications.
Washington took the first step on Friday with Pompeo's announcement.
He said any wrongdoing committed by American personnel would be dealt with in USA military and criminal courts.
Pompeo said the new visa restrictions would include "persons who take or have taken action to request or further such an investigation".
"I'm announcing a policy of USA visa restrictions on those individuals directly responsible for any ICC investigation of U.S. personnel", the secretary of state said.
The prosecution's request said there is information that members of the USA military and intelligence agencies "committed acts of torture, cruel treatment, outrages upon personal dignity, rape and sexual violence against conflict-related detainees in Afghanistan and other locations, principally in the 2003-2004 period".
The ICC and its supporters, including human rights groups that denounced Pompeo's announcement, argue that it is needed to prosecute cases when a country fails to do so or does an insufficient job of it.
The Trump administration threatened in September to block the ICC personnel from entering the United States and even prosecute them if it charges USA military members serving in Afghanistan over war crimes.
Saudi Arabia rejects call for worldwide probe of Khashoggi murder
Turkey urged Saudi Arabia to extradite the perpetrators, as well as to provide information about the location of Khashoggi's body. More than a month after his death, the Central Intelligence Agency concluded Salman ordered Khashoggi's death.
Pompeo also warned about potential economic sanctions "if the ICC does not change its course".
The United States has never been a member of the ICC.
"We will not cooperate with the ICC", Bolton said.
When president George W. Bush took office in 2001, his administration promoted and passed the American Service Members Protection Act, which sought to immunize U.S. troops from potential prosecution by the ICC.
A State Department official said the United States would allow court officials to travel for meetings to the United Nations headquarters in NY.
US National Security Advisor John Bolton unveils the Trump Administration's Africa Strategy at the Heritage Foundation in Washington, December 13, 2018.
The ICC has not yet made a decision on whether to authorize that investigation. Pompeo said Friday that more measures may come.
Pompeo's announcement came after John Bolton, President Donald Trump's national security adviser and a longtime critic of the ICC, threatened to impose sanctions on court officials in September if they continued to pursue an investigation of potential crimes by US civilians or military personnel in Afghanistan.
"The US government, where possible, takes legal action against those responsible for worldwide crimes", he added, noting that it has supported prosecution of war crimes in Rwanda, the former Yugoslavia and elsewhere.
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