Asked whether he was willing to target cultural sites, Esper told Pentagon reporters: "We will follow the laws of armed conflict."
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." He did not elaborate.
Targeting cultural sites with military action is considered a war crime under international law, including a UN Security Council resolution supported by the Trump administration in 2017 and the 1954 Hague Convention for the Protection of Cultural Property.
The US drone strike on Friday that killed Iranian military commander Qassem Soleimani has sharply escalated tensions with Iran, raising fears of all-out conflict. Washington says it killed Soleimani in self-defense, aiming to disrupt his plans to attack US personnel and interests.
Amid Iranian threats of retaliation over Soleimani's killing, Trump tweeted over the weekend that the United States had targeted 52 Iranian sites, some "at a very high level & important to Iran & the Iranian culture" if Iran struck any American or American assets in retaliation.
"They're allowed to use roadside bombs and blow up our people and we're not allowed to touch their cultural sites? It doesn't work that way," Trump said on Sunday, speaking to reporters.