Mr Trump announced that the USS Boxer, an amphibious assault ship, "took defensive action" against the Iranian drone as it was "threatening the safety of the ship and the ship's crew."
"This is the latest of many provocative and hostile actions by Iran, against vessels operating in international waters," he added.
"The United States reserves the right to defend our personnel, our facilities and interests and calls upon all nations to condemn Iran's attempts to disrupt freedom of navigation and global commerce."
A Wall Street Journal reporter who was on the USS Boxer reported that an unarmed Iranian navy Bell 212 helicopter flew alongside the Boxer, yards away from the deck, before it was chased away by a US helicopter.
The convoy of six US warships passed numerous Iranian speedboats without incident, but was then tailed by a larger Iranian warship which closed to within 500 yards of the Boxer.
An Iranian surveillance drone then came even closer and was then brought down by electronic warfare jamming.
The apparent confrontation came as Iran defended its detention Sunday of a "foreign tanker" believed to be the Panamanian-flagged vessel Riah and its 12 crew for allegedly smuggling fuel.
"The vessel was en route to deliver contraband fuel received from Iranian boats to foreign ships in farther regions when it was intercepted," Revolutionary Guards' Sepahnews website said.
Iran denied on Friday denied lost a drone in the Strait of Hormuz after President Donald Trump said that a US Navy ship had "destroyed" an Iranian drone.
Abbas Araqchi, Deputy Foreign Minister, said: "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!."
Brigadier General Abolfazl Shekarchi, an Iranian armed forces spokesman., added: Despite Trump's baseless and delusional claims, all of (Iran's) drones... have safely returned to their bases."
"There have been no reports of a confrontation with the American USS Boxer" naval vessel, he added, quoted by Tasnim news agency.
Mr Trump said on Thursday that the drone had flown to within 1,000 yards of the USS Boxer and had ignored "multiple calls to stand down" in the latest episode to stir tensions in the Gulf.
It was the first US military engagement with Iran following a series of increasingly serious incidents.