In remarks at the White House, Trump blamed Iran for a "provocative and hostile" action and said the United States responded in self-defense.
"The drone was immediately destroyed", he said, adding the drone came within about 1,000 yards of the ship and ignored "multiple calls to stand down".
Later, Trump issued a tweet about the drone takedown, reiterating that the Boxer "took defensive action" against the drone.
In June the Islamic Revolutionary Guard Corps (IRGC) announced that they had shot down a United States surveillance drone over the waters of Iran's Hormozgan province.
Trump ordered a retaliatory military strike but called it off at the last moment, saying the risk of casualties was disproportionate to the shootdown by Iran, which did not cost any USA lives.
According to Cmdr. Rebecca Rebarich, a Pentagon spokeswoman, the drone "was assessed to be Iranian". "The drone was immediately destroyed", Trump said from the East Room.
Iran's state television did not identify the seized vessel, but said it was intercepted on Sunday.
The latest actions came amid heightened tensions between the USA and Iran in the wake of the breakdown of the nuclear deal, as the secretive regime announced it exceeded the threshold of low-enriched uranium stockpile as agreed upon in the 2015 accord.
The recent incidents in the Strait of Hormuz, which have sharpened fears of a potential military conflict between the two countries, occurred against the backdrop of the fraying Iranian nuclear deal.
Pressure in the region has been building since the Trump administration withdrew from the 2015 nuclear agreement and imposed punishing sanctions on the country.
The state media later published footage of Iranian speedboats circling around the Panamanian-flagged Riah tanker.
The Guards said the impounded ship was in the area of Larak Island in the Gulf and had 12 foreign crew.
The announcement came just days after an oil tanker based in the United Arab Emirates disappeared off trackers in Iranian territorial waters.
"We do this (inspecting ships) every day". 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. They later fell, however, on weakness in US equities markets and an expectation that crude output would rise in the Gulf of Mexico following last week's hurricane in the region. While Iran has been blamed for attacks on merchant shipping, it has denied responsibility.
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