Trump said the Iranian drone came about 1,000 yards away from the USS Boxer and ignored calls to stand down.
Iran downed a USA drone in June that it says violated its airspace - a claim contested by the Trump administration which says it remained above global waters.
Iran shot down a US drone last month that it said was flying over its territory. Mr Trump called off a planned retaliatory air strike at the last minute. While this tanker and drone tit-for-tat has been going on, the U.S is working behind the scenes to try to avoid a military conflict with Iran.
On Thursday, the Boxer transited through the Strait of Hormuz, a strategic waterway where one fifth of the world's oil passes.
"The drone was immediately destroyed", Trump continued.
Iran played down the seizure of the ship, which it said was a small vessel that was smuggling oil.
Iran reduces commitments to nuclear deal to give chance to diplomacy
Iran has threatened to abandon its nuclear commitments unless European countries do more to help its sanction-hit economy. Trump's remark came after the U.S. had accused Tehran of two separate attacks on oil tankers in the Persian Gulf .
With tensions between the USA and Iran still high, Iran's paramilitary Islamic Revolutionary Guard Corps has seized a foreign oil tanker, accusing the crew of smuggling oil.
A Central Command spokesman was unable to say how the Boxer took the drone down, but CNN reporter Ryan Browne said in a tweet that U.S. Marines aboard the Boxer brought it down using jamming equipment.
"We do this (inspecting ships) every day".
Washington has recently beefed up its military presence and the U.S. Fifth Fleet, based in Bahrain, says that Gulf Arab states have stepped up patrols.
Iran released a video of the seized tanker, which matches the Riah, Haaretz reported. If, on the other hand, talks with Iran go well and they are allowed to sell their oil, then the prices for the commodity would more than likely head back to the $40 level.
Also Thursday, the USA sanctioned a number of companies and individuals it said were involved in an illicit network to procure materials for Iran's nuclear program.
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