IS smashed priceless artefacts at the Mosul museum, then bulldozed the city of Nimrud, which was founded in the 13th century BC.
The jihadists may now have turned their attention to the extremely well preserved fortress city Hatra, which is more than 2,000 years old and a UNESCO world heritage site, with the United Nations condemning its reported "destruction".
"The sky is not in the hands of the Iraqis, the sky is not in our hands. Therefore, the international community must move with the means it has," Adel Fahad al-Shershab told journalists in Baghdad.
Asked specifically if he wanted coalition strikes to protect archaeological sites, he responded: "What I request from the international community and the international coalition is to carry out air strikes against terrorism wherever it is found."
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The attacks on Iraq's archaeological heritage took place in IS-held areas in the northern province of Nineveh, where Iraq does not have security forces that are able to respond on the ground.
Targeting militants destroying archaeological sites would be a departure for the coalition, which is waging a campaign of strikes aimed at weakening IS military capabilities in Iraq and neighbouring Syria.
"The site of Hatra is a site in the desert where it is possible to see any infiltration" from the air, Shershab said of the ancient city, which features a unique blend of eastern and western architecture.
"It was expected that they (IS) would destroy it," he said.
But it remains unclear whether large-scale destruction was carried out at Hatra, whose thick walls and large buildings withstood two Roman invasions in the 2nd century AD.
The jihadists spearheaded a sweeping offensive last June that overran large areas north and west of Baghdad, and Iraqi forces backed by the US-led coalition and Iran are battling to push them back.
IS tries to justify the destruction of antiquities by saying they are idolatrous, but experts say the group traffics in them to fund its self-proclaimed "caliphate" and destroys only those pieces that are too bulky to be smuggled.