The USS Boxer "took defensive action" against the Iranian drone as it was "threatening the safety of the ship and the ship's crew," Trump announced at the White House.
"The drone was immediately destroyed," he added.
The downing of the drone comes at a time of mounting tensions in the Gulf region between the US and Iran which has been slapped with a raft of sanctions by the Trump administration.
"This is the latest of many provocative and hostile actions by Iran, against vessels operating in international waters," said Trump.
"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.
"I also call on other nations to protect their ships as they go through the Strait and to work with us in the future."
Iran's foreign minister, Mohammad Javad Zarif, said on Thursday that Iran and the US were only "a few minutes away from a war" after Iran downed an American drone.
He spoke to US-based media on the sidelines of a visit to the United Nations.
Earlier Thursday, Iran said its Revolutionary Guard seized a foreign oil tanker and its crew of 12 for smuggling fuel out of the country, and hours later released video showing the vessel to be a UAE-based ship that had vanished in Iranian waters over the weekend.
The announcement cleared up the fate of the missing ship but raised a host of other questions and heightened worries about the free flow of traffic in the Strait of Hormuz, one of the world's most critical petroleum shipping routes.
One-fifth of global crude exports passes through the strait.
