Updated at 4:08 p.m. ET
President Trump said that Iran appears "to be standing down" after Tuesday night's missile attack in Iraq and that "the American people should be extremely grateful and happy no Americans were harmed."
Trump, in a nationally televised address from the White House, also announced a new round of what he termed "punishing economic sanctions" against the Iran government. And he called on NATO to become "much more involved in the Middle East process."
Later in the day, Trump spoke by phone with NATO Secretary General Jens Stoltenberg. According to a White House statement, "The President emphasized the value of NATO increasing its role in preventing conflict and preserving peace in the Middle East."
Trump credited the dispersal of forces at the base "and an early warning system that worked very well" for the lack of casualties.
Soleimani "should have been terminated long ago," Trump said, calling him "the world's top terrorist."
Defense Secretary Mark Esper suggested Tuesday that Soleimani had been a matter of days away from launching more violence against Americans or U.S. interests, but public details remain scarce.
Trump blamed the Obama administration, which reached an agreement limiting Iran's ability to pursue nuclear weapons in return for — among other things — millions of dollars of Iranian assets that had been held by the U.S., for the Tuesday night missile strike.
"The missiles fired last night at us and our allies were paid for with the funds made available by the last administration," he charged.
Trump withdrew from that 2015 agreement, and on Wednesday, he called on the nation's European allies along with Russia and China to do so as well.
Asked about Trump's allegation, former Obama Defense Secretary Leon Panetta told Here & Now, "I think the president, when he's caught in a corner, always winds up blaming President Obama for whatever's happening, but the reality is that this president does not have a clear or comprehensive strategy in dealing with the Middle East. It's obvious from his own remarks."
Late Tuesday night, Trump acknowledged the attacks by tweeting, "All is well!"
"Assessment of casualties & damages taking place now. So far, so good!" he added.
Initial satellite images of Ain al-Assad air base, in western Iraq, suggested that Iran's strike targeted aircraft repair shelters and other such facilities.
It wasn't immediately clear what effect the damage might have on American military capabilities in Iraq, but the United States has other air bases and vast forces arrayed around the Middle East.
A defense official told reporters in a statement that U.S. early warning systems "detected the incoming ballistic missiles well in advance, providing US and Coalition forces adequate time to take appropriate force protection measures."
The official said the Defense Department knew of the attack before U.S. military officials were alerted by Iraqi officials, who were reportedly notified of the impending attacks by Iran.
Mohammad Javad Zarif, the Iranian foreign minister, called Tuesday night's missile strike "proportionate measures in self-defense."
"We do not seek escalation or war, but will defend ourselves against any aggression," he said on Twitter.
Before the attacks, he also told NPR's Mary Louise Kelly, who is in Tehran, that Soleimani's killing "amounts to war."
Wednesday's address followed days of tensions that began last week as Iran-backed Iraqi militiamen and their supporters targeted the U.S. Embassy in Baghdad, setting fires and damaging a perimeter gatehouse at the complex.
In the days after the Soleimani strike, the Trump administration faced domestic political criticism over the lack of transparency following the operation. The administration maintained that the U.S. faced an imminent threat, however.
"If you're looking for imminence, you need look no further than the days that led up to the strike," U.S. Secretary of State Mike Pompeo said Tuesday, alluding to the violent protests outside the U.S. Embassy.
Members of Congress who were briefed Wednesday by Trump administration officials about the killing of Soleimani had mixed responses, reflecting partisan divisions.
Trump supporter Rep. Mark Meadows, R-N.C., said, "It leaves little doubt in my mind — and certainly should leave little doubt in any member's mind — that not only did the president make the right call, but that this was a clear and present danger for American interests and American individuals."
California Democratic Rep. Barbara Lee said in contrast insisted that Congress should reassert "its role in exercising our constitutional responsibility."
"If in fact any president wants to use force or conduct military strikes, then they must, he or she must, come to Congress for that authorization," she said.
The House is expected to vote this week on a measure that would limit Trump's abilities to order a military strike against Iran.
Relations between the U.S. and Iran have been fraught since the 1979 Islamic Revolution, but they have deteriorated sharply since the Trump administration withdrew from the nuclear deal.
The Trump administration maintained that the agreement with the world's powers gave Iran too much in exchange for too little. The administration instead imposed harsh sanctions designed to squeeze Iran's already moribund economy and vowed to impose "maximum pressure."
Since Soleimani's killing, Iran has said it will no longer abide by the restrictions imposed on its nuclear program by the deal but added that its decision is reversible if the other signatories to the accord — Britain, China, France, Germany, Russia and the European Union — provide it economic benefits.
National correspondent Tom Bowman contributed to this report.