Mr Pompeo issued the threat after a series of rocket attacks on US military believed to have been carried out by militia groups with close links to the Islamic Republic’s regime. The latest attacks saw Katyusha rockets land near US military positions on the perimeter of Baghdad International Airport.
Mr Pompeo said: “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.
“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.”
The US 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 Shi’ite paramilitary groups.
Two rocket attacks this week targeted a compound near Baghdad International Airport, which houses US troops, with strikes last Monday wounding Iraqi troops.
Mr Pompeo said: “We hope and pray these brave Iraqis will quickly and fully recover from their injuries.”
The Secretary of State, who has repeatedly warned Tehran, blamed the latest attacks squarely on “Iran’s proxies”.
READ MORE:Iran behind increasingly sophisticated rocket attacks on US bases
A senior US military official said the attacks were becoming more frequent, more sophisticated and were jeopardising the coalition’s ability to effectively combat ISIS fighters still operating in the strife-torn region.
He said: “We’re used to harassing fire but the pace of that was previously pretty episodic.
“Now the level of complexity is increasing, the volume of rockets being shot in a single volley is increasing and is very concerning to us.
“There is a point at which their actions change things on the ground and make it more likely that some other actions, some other choices made – by somebody, whether it’s them or us – will escalate unintentionally.”
The military official said Iranian-armed militias were approaching a red line where the coalition would respond with force, and “no one will like the outcome”.
There have been no claims of responsibility for any of the attacks.
But the US military official said intelligence and forensic analyses of the rockets and launchers pointed to Iranian-backed Shi’ite Muslim militia groups, notably Kataib Hezbollah and Asaib Ahl al-Haq (AAH).
The US military official said Iraq’s Shi’ite-led government had not taken action over the incidents.
He said: “It’s very concerning to me. That it’s acceptable that we’re getting attacked by elements that are supposedly being brought in under the heel of the Iraqi government as part of its security forces.”
Iran has gained overwhelming influence in neighbouring Iraq, with which it shares a Shi’ite majority, since the 2003 US invasion brought down Saddam Hussein.
Donald Trump’s administration, which is close to Saudi Arabia and Israel, has been trying to block Tehran’s influence in the region through heavy economic sanctions.