He said the USS Boxer took “defensive action” after the drone ignored repeated warnings and came within 1,000 yards of the ship.
The president called it a “provocative and hostile” action by Iran.
The “fixed-wing unmanned aerial system… closed within a threatening range” at 10am local time on Thursday, according to a Pentagon statement.
“The ship took defensive action against the UAS to ensure the safety of the ship and its crew,” it added.
The incident comes amid a period of heightened tension between the two countries, which included Iran last month shooting down a US drone it said was flying over the country.
That incident saw Mr Trump call off a retaliatory strike at the last minute because he said too many people would have died.
The Strait of Hormuz is a vital shipping lane for oil and gas, but has become a flash point following the US decision to pull out of a nuclear deal with Iran and impose strict sanctions that are seriously damaging Iran’s economy.
A foreign tanker with 12 crew was seized by Iran in the strait on Sunday, according to state television in the country.
It claimed the Panamanian-flagged MT Riah was smuggling around one million litres of fuel from Iranian smugglers to foreign customers.
Iran has recently increased uranium production and enrichment over the limits of the 2015 nuclear deal to try to put pressure on Europe to offer it better terms and allow it to sell its oil abroad.
The US has subsequently sent thousands of additional troops, nuclear-capable B-52 bombers and fighter jets to the region.
Western powers have also blamed Iran for two mine attacks on oil tankers last month off the country’s coast – something it denies.
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