Lawmakers have criticized Mr. Trump for not telling them in advance of the strike on General Suleimani. Among those attacking Mr. Trump’s actions is a Republican, Senator Rand Paul of Kentucky, who wrote on Twitter that the main question “is whether the assassination of Soleimani will expand the war to endanger the lives of every American soldier or diplomat in the Middle East?”

When asked Sunday by several interviewers whether the United States would attack cultural sites, Mr. Pompeo avoided answering directly. He said on ABC’s “This Week” that the United States would “behave lawfully” and “behave inside the system.”

In one appearance, Mr. Pompeo said the costs of any attacks by Iran or its partners on American interests would “be borne by Iran and its leadership itself.”

Mr. Pompeo made his statements as Iraq’s Parliament voted to expel more than 5,000 American troops from Iraq. The United States military has been fighting the Islamic State, a Sunni militant group that seized enormous swaths of territory in Iraq and Syria in 2014 but is now transforming into an insurgency after years of military defeats.

Prime Minister Adel Abdul Mahdi of Iraq, a Shiite leader who has ties to Tehran, has criticized the presence of American troops following the strike on General Suleimani, though he has not yet signed Parliament’s bill.

American military officials said Sunday that they were suspending operations against the Islamic State as American forces braced for retaliation by Iran and its partners.

Asked about the Iraqi Parliament’s vote, Mr. Pompeo said on the CBS program “Face the Nation” that the United States would continue to battle the Islamic State. “It is the United States that is prepared to help the Iraqi people get what it is they deserve and continue our mission there to take down terrorism from ISIS and others in the region,” he said.