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However, he told reporters it could have been the result of human error, saying: "I find it hard to believe it was intentional."
Iran said the drone had violated Iranian airspace, but the US military denied this.
The incident comes amid escalating tension between the two countries.
Iran's Foreign Minister Javad Zarif said Iran would take its complaint that the US "encroaches on our territory" to the UN.
We don't seek war but will zealously defend our skies, land and waters," he said on Twitter.
Speaking at the White House, he called the drone's downing a "new fly in the ointment".
Mr Trump said it was "documented" that the unmanned drone had been over international waters and not in Iranian airspace.
"I think probably Iran made a mistake - I would imagine it was a general or somebody that made a mistake in shooting that drone down," he said.
"It could have been somebody who was loose and stupid," he added.
Russia's President Vladimir Putin has warned that war between the US and Iran would be a "catastrophe with unpredictable consequences".
UN Secretary General Antonio Guterres has urged all parties to exercise maximum restraint.
In the US, Democratic House Speaker Nancy Pelosi said the US had no appetite for war with Iran, while the leading candidate for the Democratic presidential nomination, Joe Biden, called Mr Trump's Iran strategy a "self-inflicted disaster".
Meanwhile Saudi Foreign Minister Adel al-Jubeir told the BBC his country was trying to send a message to Iran that its behaviour was "not acceptable".
"Nobody wants to start a war. But we can't let Iran go on a rampage like this. The evidence of Iranian involvement is very compelling. They said they would do it, and now they are doing it," he said.
Oil prices have jumped by about 5% following the incident.
Iran's Islamic Revolutionary Guard Corps (IRGC) said its air force had shot down a US "spy" drone in the early hours after the unmanned aircraft violated Iranian airspace near Kuhmobarak in the southern province of Hormozgan.
In a speech on Iranian state TV, IRGC commander-in-chief Maj-Gen Hossein Salami said the drone's downing was a "clear message" to the US that Iran's borders were "our red line".
Foreign minister Javad Zarif later said Iran had retrieved sections of the US military drone "in our territorial waters where it was shot down". He added that the aircraft had taken off from the UAE in "stealth mode".
Iran released unverified footage of what it said was the US drone being destroyed.
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