WASHINGTON — The U.S. military may strike more Iranian leaders if the Islamic Republic retaliates for the Trump administration’s killing of Tehran’s most powerful general last week by attacking Americans or American interests, Secretary of State Mike Pompeo said Sunday.

As Pompeo conducted a round of TV interviews to explain President Donald Trump’s decision to target Iranian Gen. Qassem Soleimani, the repercussions from that attack played out: The Iraqi Parliament called on the 5,200 U.S. forces in the country to leave; the U.S. military coalition in Baghdad suspended training of Iraqi forces to concentrate on defending coalition troops; and in Beirut, the Lebanese Hezbollah chief said U.S. forces throughout the Mideast are fair targets for retaliation.

Even a Trump ally, Sen. Lindsey Graham, R-S.C., called the move by Iraqi lawmakers “a bit concerning.’’

In Tehran, Iranian state television reported that the country will no longer abide by any limits of the 2015 nuclear deal it signed with the United States and other world powers. Trump withdrew the U.S. from the deal in 2018 and stepped up economic sanctions on Tehran — actions that accelerated a cycle of hostilities leading to the Soleimani killing.

The State Department had no immediate comment on Iran reportedly abandoning the nuclear deal, a move that holds the prospect of Iran accelerating its production of materials for a nuclear weapon.

Trump continued issuing warnings to Iran by tweet. “These Media Posts will serve as notification to the United States Congress that should Iran strike any U.S. person or target, the United States will quickly & fully strike back, & perhaps in a disproportionate manner,” he wrote Sunday afternoon. “Such legal notice is not required, but is given nevertheless!”

Democrats in Congress complained about the administration’s failure to consult with legislative leaders before conducting the drone attack Friday against Soleimani, and the White House faced a barrage of questions about the killing’s legality. Pompeo said the administration would have been “culpably negligent” in its duty to protect the United States if it had not killed Soleimani, although he did not provide evidence for his previous claims that Soleimani was plotting imminent attacks on Americans.

Instead of arguing that an attack had been imminent, Pompeo said it was inevitable.

“We watched him continue to actively build out for what was going to be a significant attack – that’s what we believed – and we made the right decision,” he said, adding later: “We continue to prepare for whatever it is the Iranian regime may put in front of us within the next 10 minutes, within the next 10 days, and within the next 10 weeks.”

Congressional Democrats were skeptical.

“I really worry that the actions the president took will get us into what he calls another endless war in the Middle East. He promised we wouldn’t have that,” said Chuck Schumer of New York, the Senate’s top Democrat.

Schumer said Trump lacks the authority to engage militarily with Iran and Congress needs a new war powers resolution “to be a check on this president.” To which Pompeo said: “We have all the authority we need to do what we’ve done to date.”

Sen. Mark Warner, D-Va., said the administration violated the Constitution by not consulting with Congress in advance.

“It’s also important because one, you potentially get members of Congress to buy in ahead of time, and two, they may ask that hard question that’s not asked in an insular group,” Warner said.

Congressional staffs got their first briefings from the administration on Friday, and members are expected to be briefed this week.

Pompeo appeared on six news shows while Trump kept silent on the final day of his holiday break in Florida. The appearances by the top American diplomat appeared aimed at dissuading Iran from launching a major retaliation for the Soleimani killing. The Iranians have said the U.S. should expect a strong response. They have a range of options, from cyberattacks to military assaults.

It was unclear whether the administration would attempt a back door communication with Iran in pursuit of its stated goal of “de-escalation” of tensions. Retired Gen. David Petraeus, an ex-CIA director and former commander of U.S. forces in Iraq, said he believes the administration needs a strategy for tamping down the chances of all-out war.

“I think the real question for the United States is, will there be a diplomatic initiative that says, OK, look, this is not headed in a good direction. We truly do want to de-escalate. Everyone is going to lose if this continues to ratchet upward. Can we now sit down and talk,” Petraeus said.

Pompeo declined to say whether he had sought to communicate with Iran since Friday. He stressed the U.S. resolve to hold Iran accountable for its interventions in Iraq, Syria and elsewhere in the Mideast.

Pompeo said the Obama administration had tried to “challenge and attack everybody who was running around with an AK-47 or a piece of indirect artillery. We’ve made a very different approach. We’ve told the Iranian regime, ‘Enough. You can’t get away with using proxy forces and think your homeland will be safe and secure.’ We’re going to respond against the actual decision-makers, the people who are causing this threat from the Islamic Republic of Iran.”

He said the cost to Iran if it uses proxy forces to hit American targets will come down on not just those proxies, which are present in Iraq, Syria, Yemen, Lebanon and elsewhere.

“They will be borne by Iran and its leadership itself,” Pompeo said. “Those are important things the Iranian leadership needs to put in its calculus as it makes its next decision.”

Pompeo tip-toed around questions about Trump’s tweet Saturday threatening to attack Iranian cultural sites, a military action that likely would be illegal under the laws of armed conflict and the U.N. charter.

Trump wrote that if Iran were to strike “any Americans, or American assets, we have targeted 52 Iranian sites (representing the 52 American hostages taken by Iran many years ago), some at a very high level & important to Iran & Iranian culture, and those targets, and Iran itself, WILL BE HIT VERY FAST AND VERY HARD.”

Pompeo said any U.S. military strikes inside Iran would be legal.

“We’ll behave inside the system,” Pompeo said. “We always have and we always will.”

One U.S. national security official said Trump’s threat had caught many in the administration off guard and prompted calls for others in the government, including Pompeo, to clarify the matter. The official, who was not authorized to speak publicly to the issue, said clarification was necessary to affirm that the U.S. military would not intentionally commit war crimes.

Oona Hathaway, an international law professor at Yale and a former national security law official in the Defense Department’s legal office, said in an interview that Trump’s threat amounted to “a pretty clear promise of commission of a war crime.” She said the Soleimani killing likely also was illegal because the administration has not shown the threat he posed was imminent in the sense that it was so urgent that action was required without consulting Congress.

She cited legal problems with both of Trump’s Saturday threats — the threat to hit 52 targets in Iran for symbolic reasons, and the threat to strike Iranian cultural sites. Both, she said, would be war crimes — the targeting of 52 sites because Trump justifies it on symbolic grounds of retribution for Iran’s hostage-taking 40 years, and the hitting of cultural sites because that would be illegal under the 1954 Hague Convention for the protection of cultural sites.

The Pentagon’s chief spokesman, Jonathan Hoffman, referred questions about the Trump tweet to the White House.

Some of the Democrats running to challenge Trump in November questioned whether he had a long-term plan for the Mideast.

“When you’re dealing with the Middle East, you need to think about the next and the next and the next move. This is not checkers. And I’m not sure any of us really believe that this president and the people around him” are “really going through all of the consequences of what could happen next,” said Pete Buttigieg, the former mayor of South Bend, Indiana.

And Massachusetts Sen. Elizabeth Warren, a juror in Trump’s upcoming impeachment trial, wondered about the timing of the attack on the Iranian general.

“I think the question people reasonably ask is, ‘Next week Donald Trump faces the start potentially of an impeachment trial. And why now?’ I think people are starting to ask, ‘Why now did he do this? Why not delay?’ And why this one is so dangerous is that he is truly taking us right to the edge of war,” she said.

Pompeo appeared on ABC’s “This Week,” CNN’s “State of the Union,” NBC’s “Meet the Press,”’ CBS’ “Face the Nation,’’ “Fox News Sunday” and Fox News Channel’s “Sunday Morning Futures.” Schumer was on ABC, Warner and Warren were on NBC, Petraeus was on CBS, Buttigieg was on CNN and Graham was on Fox News Channel.