"We must... use this opportunity to remind Iran's leaders that any attacks by them, or their proxies of any identity, that harm Americans, our allies or our interests will be answered with a decisive US response," Pompeo said in a statement.
"Iran must respect the sovereignty of its neighbours and immediately cease its provision of lethal aid and support to third parties in Iraq and throughout the region," he said.
Rockets hit US airbase in northern Iraq
The United States has expressed mounting concern about the flurry of attacks on Iraqi bases used by US troops, several of which it has blamed on Iranian-backed Shia paramilitary groups.
Two rocket attacks this week targeted a compound near Baghdad International Airport, which houses US troops, with an incident Monday wounding Iraqi troops.
"We hope and pray these brave Iraqis will quickly and fully recover from their injuries," Pompeo said.
Pompeo, who has repeatedly warned Tehran, pinned the blame for the latest attacks squarely on "Iran's proxies."
Iran has gained overwhelming influence in Iraq, its neighbour with which it shares a Shia majority, since the 2003 US invasion brought down Saddam Hussein.
President Donald Trump's administration, which is close to Iran's adversaries Saudi Arabia and Israel, has been trying to counter Tehran's influence around the region, including through sanctions aimed at blocking all its oil exports.
Iraq’s embattled PM bows out as protesters mourn dead
The United States last week imposed sanctions on three leaders of the Shia paramilitary force Hashed al-Shaabi, accusing them of taking part in the deadly crackdown on nationwide protests.
Iraqi prime minister Adel Abdel Mahdi, a close ally of Iran who also enjoyed cordial relations with the United States, resigned after the two months of demonstrations in which around 460 people have died.
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