Speaking at the White House, Trump said that USS Boxer an amphibious assault ship of the US Navy "took defensive action" on Thursday as the Iranian drone was "threatening the safety of the ship and the ship's crew".
"The drone was immediately destroyed," the US President added without giving more details.
The incident marks a new escalation of tensions between the two countries less than a month after Iran shot down an American drone in the same waterway and Trump came close to retaliating with a military strike.
Reacting to Trump's statement, Iranian Deputy Foreign Minister Abbas Araqchi said that Iran had not lost any drone in the Strait of Hormuz, one of the world's most critical petroleum shipping routes as one-fifth of global crude exports pass through it.
"We have not lost any drone in the Strait of Hormuz nor anywhere else. I am worried that USS Boxer has shot down their own UAS (Unmanned Aerial System) by mistake!," Araqchi tweeted, referring to a US warship in the strategic waterway.
The development comes after Tehran on Thursday said that it had seized a "foreign tanker" and its 12 crew for "smuggling fuel" in the Gulf and days after reports of a British tanker being harassed in the region.
Chief Pentagon spokesperson Jonathan Hoffman said that USS Boxer downed the approaching drone at approximately 10 a.m. on Thursday during its planned inbound transit of the Strait of Hormuz, the US media reported.
"A fixed wing 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 had earlier told reporters in New York that "we have no information about losing a drone...".
Tensions are running high between Washington and Tehran over Trump's decision to pull the US out of the 2015 nuclear deal which Iran signed with world powers. The US President reinstated sanctions on Iran following the withdrawal.
"This is the latest of many provocative and hostile actions by Iran against vessels operating in international waters. The US reserves the right to defend our personnel facilities and interests and calls upon all nations to condemn Iran's freedom of navigation...," said Trump.
"I also call on other nations to protect their ships as they go through the strait and to work with us in the future," he added.
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, claiming that these measures are "prudent response to credible threats from Iran" and for "defensive purposes".
( With inputs from IANS )