Trump scolded both Iran and Israel for early violations of the truce that he had announced but directed particularly stinging criticism at Israel over the scale of its strikes, telling it to "calm down now".
He said Israel called off further attacks at his command to preserve the deal to end a 12-day air war with Iran, the biggest ever military confrontation between the Middle East arch-foes that had raised fears of global repercussions.
Following those remarks, however, two witnesses reached by telephone in the Iranian capital said they heard two loud blasts.
Prime Minister Benjamin Netanyahu's office acknowledged that Israel had bombed a radar site near Tehran in what it said was retaliation for Iranian missiles fired three-and-a-half hours after the ceasefire had been due to begin.
It said Israel had decided to refrain from further attacks following a call between Netanyahu and Trump but did not explicitly say whether the strike on the radar site took place before or after they spoke.
Iran denied launching any missiles and said Israel's attacks had continued for an hour-and-a-half beyond the time the truce was meant to start.
Iranian President Masoud Pezeshkian said later that Iran would not violate the ceasefire unless Israel did so, and that it was prepared to return to the negotiating table, according to state-run Nournews.
Trump, en route to a NATO summit in Europe, had admonished Israel with an obscenity in an extraordinary early morning outburst at an ally whose air war he had joined two days before by dropping massive bunker-buster bombs on Iran's underground nuclear sites.
"All planes will turn around and head home, while doing a friendly 'Plane Wave' to Iran. Nobody will be hurt, the Ceasefire is in effect!" Trump said in a post on Truth Social.
That followed a post in which he had said: "Israel. Do not drop those bombs. If you do it it is a major violation. Bring your pilots home, now!"
Before departing the White House, Trump told reporters he was unhappy with both sides for breaching the ceasefire but particularly frustrated with Israel, which he said had "unloaded" shortly after agreeing to the deal.
"I've got to get Israel to calm down now," Trump said.
Iran and Israel had been fighting "so long and so hard that they don't know what the f*** they're doing".
A reporter for Axios said Netanyahu had told Trump that Israel would scale back the bombing mission rather than cancel it.
Despite the initial reports of violations, in both countries there was a palpable sense of relief that a path out of war had been charted, 12 days after Israel launched it with a surprise attack, and two days after Trump joined in with strikes on Iranian nuclear targets.
"We're happy, very happy. Who mediated or how it happened doesn't matter. The war is over. It never should have started in the first place," said Reza Sharifi, 38, heading back to Tehran from Rasht on the Caspian Sea, where he had relocated with his family to escape the air strikes.
Arik Daimant, a software engineer in Tel Aviv, said: "Regrettably, it's a bit too late for me and my family, because our house back here was totally destroyed in the recent bombings last Sunday. But as they say: 'better late than never', and I hope this ceasefire is a new beginning."
Speaking to reporters on board Air Force One heading to the NATO summit, Trump said he did not want to see Iran's ruling system toppled.
"I don't want it. I'd like to see everything calm down as quickly as possible. Regime change takes chaos and ideally we don't want to see so much chaos," he said.
The ceasefire came the morning after Iran responded to the US participation in the air strikes by firing a volley of missiles at the biggest U.S. military base in the Middle East, located in Qatar.
No one was hurt in that strike as the Iranian retaliation appeared to have been calibrated carefully to allow de-escalation.Â
Trump thanked Iran for warning the United States in advance to avoid injuries.
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