Speaking at the White House on Thursday, Trump said that USS Boxer, an amphibious assault ship of the US Navy, destroyed the drone that threatened the US warship by flying within 1,000 yards of it and ignored multiple warning, Xinhua reported.
Trump said the drone was "threatening the safety of the ship and the ship's crew," calling the downing a "defensive action".
Chief Pentagon spokesperson Jonathan Hoffman said later in a statement that USS Boxer downed the approaching drone at approximately 10 a.m. during its planned inbound transit of the Strait of Hormuz.
"A fixed wing unmanned aerial system (UAS) approached USS Boxer and closed within a threatening range. The ship took defensive action against the UAS to ensure the safety of the ship and its crew," said the statement.
Citing US defence officials, the US media reported that the Iranian drone was brought down by counter-drone jamming equipment, which was operated by US marines aboard the USS Boxer.
Iranian Foreign Minister Mohammad Javad Zarif later told reporters at the UN headquarters: "We have no information about losing a drone today."
The incident came as Iran's Islamic Revolutionary Guard Corps (IRGC) announced that it had seized a foreign ship "smuggling fuel" in the Gulf on Thursday and days after reports of a British tanker being harassed in the region.
The downing of the Iranian drone also came one month after a US drone operating over the Strait of Hormuz was shot down by Iranian air defence system on June 19, which almost led to a US military retaliation against Iran.
USS Boxer, a Wasp-class amphibious assault ship, arrived in US Central Command area of operation on June 23 with the 11th Marine Expeditionary Unit and two other amphibious warships, according to a statement of US 5th Fleet.
The US has been upgrading its military posture in the Middle East since May by sending troops and military assets to the region. Pentagon claimed these measures are "prudent response to credible threats from Iran" and for "defensive purpose".
"Anybody who starts a war with Iran will not be the one who ends it," Zarif warned on Wednesday, referring to the escalating tensions between Iran and the US amid US military buildup in the region.
( With inputs from IANS )