Soleimani's daughter, Zeinab, directly threatened the US military in the Middle East while also warning President Donald Trump, whom she called "crazy".
Pressed on whether he would then not target such sites, because that would be a war crime, Esper said,"That's the laws of armed conflict".
The split between the president and his Pentagon chief came amid heightened tensions with Tehran following a USA drone strike that killed Gen. Qassem Soleimani, the head of Iran's elite Quds Force. Those targets, he added, would be hit "very fast and very hard" if Tehran retaliated for the killing of top general Qassem Soleimani.
The wider region has suffered many cultural attacks carried out by the Islamic State group, which targeted mosques, shrines, churches and famous sites such as Palmyra in Syria.
But Iran's pronouncement said Tehran's cooperation with the International Atomic Energy Agency (IAEA), which has been monitoring the country's nuclear activities and sites, will continue as before.
In an apparent reaction to that message, Trump tweeted: "IRAN WILL NEVER HAVE A NUCLEAR WEAPON!"
"They're allowed to kill our people, " he told reporters.
"They fear that someone comes to the USA and reveals realities", Zarif said and added: "The world is not limited to NY and you can talk to the American people from Tehran and we will do that". And we're not allowed to touch their cultural sites? "It doesn't work that way".
"He doesn't know worldwide law".
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The Geneva Conventions prohibit "acts of hostility directed against the historic monuments, works of art or places of worship which constitute the cultural or spiritual heritage of peoples".
Iran has vowed revenge and threatened both the USA and Israel. Maj.
General Soleimani, who commanded the elite Quds Force, was responsible for building up Iran's network of proxy armies across the Middle East. Those targets and war plans are routinely updated, including during the recent uptick in hostilities.
Iran's announcement increases the threat posed by the country's nuclear program, Kelsey Davenport, director for Nonproliferation Policy at the Arms Control Association, told RFE/RL. Only after that process is complete would a list of potential sites go before the president for approval. The U.S. continued to reinforce its own positions in the region and warned of an unspecified threat to shipping from Iran in the region's waterways, key routes for global energy supplies.
"It shows that he is somewhat deranged about this", said Senator Tim Kaine.
The alliance said it encourages members and advocates to contact their legislators to protest the threat to attack cultural sites. "We can say America, Mr. Trump, has taken action directly against us - so we take direct action against America".
Trump broke precedent by failing to inform congressional leaders before the attack on Soleimani, and by making classified his formal notification to Congress of the attack on Saturday.
"End of malign US presence in West Asia has begun", Zarif asserted. The U.S. could not win that war, but neither could Iran, and it would suffer huge damage if there were a flat-out American bombing campaign using only conventional bombs and warheads.
"We're here today to pay respects to the great commander of the holy defence", said one of the mourners who came from the southern city of Shiraz to attend the funeral in Kerman.