Secretary of State Mike Pompeo ordered that all but emergency staff leave Basra, with consular duties to be taken over by the embassy in Baghdad.

Basra has been rocked by weeks of demonstrations by Iraqis protesting unemployment and government inefficiency in the oil-rich south.

Pompeo, who has made rolling back Iranian influence in the region a top priority, blamed militias linked to Iran's elite Revolutionary Guards for "indirect fire" -- which usually means rockets or artillery -- against the US consulate.

"I have advised the government of Iran that the United States will hold Iran directly responsible for any harm to Americans or to our diplomatic facilities in Iraq or elsewhere and whether perpetrated by Iranian forces directly or by associated proxy militias," he said in a statement while in New York to attend the UN General Assembly.

"I have made clear that Iran should understand that the United States will respond promptly and appropriately to any such attacks," he added.

Ruled by Shiite clerics, Iran has a strong influence in Iraq, especially in the country's Shiite-majority south.

Iran's consulate itself was earlier burned down during the protests, although no one was injured and Tehran quickly opened a new mission.

Iran's foreign ministry earlier called US allegations of fomenting violence in Basra "astonishing, provocative and irresponsible."

Speaking at the General Assembly, Iranian President Hassan Rouhani also dismissed the US criticism of involvement in its neighbor, questioning why Washington itself was involved in a country so far away.

Earlier this month three mortar rounds also hit the fortified Green Zone area in Baghdad, home to the US embassy, without causing injuries and with the perpetrators unclear.

Diplomatic security is a key priority for the United States and Pompeo, who as a congressman went on the offensive against former secretary of state Hillary Clinton over a deadly attack on the US consulate in the eastern Libyan city of Benghazi.