He called on other countries to condemn what he says are Iran’s attempts to disrupt the freedom of navigation and global commerce in the strategic waterway in the Persian Gulf.
Mr Trump said the USS Boxer took defensive action after the drone closed to within 1,000 yards of the warship and ignored multiple calls to stand down.
The president said the drone threatened the safety of the American ship and its crew.
“The United States reserves the right to defend our personnel, facilities and interests and calls upon all nations to condemn Iran’s attempts to disrupt freedom of navigation and global commerce,” Mr Trump said.
The Pentagon said the incident happened at 10am local time on Thursday in international waters while the Boxer was crossing the waterway to enter the Persian Gulf.
“A fixed-wing unmanned aerial system approached Boxer and closed within a threatening range,” chief Pentagon spokesman Jonathan Hoffman said in a written statement.
“The ship took defensive action against the UAS to ensure the safety of the ship and its crew.”
It was the latest in a string of moves by the US and Iran that have brought them closer to armed conflict since early May when Washington accused Tehran of threatening US forces and interests in Iraq and in the Gulf.
In response, the US accelerated the deployment of an aircraft carrier strike group to the Arabian Sea and deployed four B-52 long-range bombers to the Gulf state of Qatar. It also has deployed additional Patriot air defence missile batteries in the Gulf region.
Shortly after Iran shot down a US Navy drone aircraft on June 20, Mr Trump ordered a retaliatory military strike but called it off at the last moment, saying the risk of casualties was disproportionate to the downing by Iran, which did not cost any US lives.
Iran claimed the US drone violated its airspace but the Pentagon denied this.
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 the American drone. He spoke to US-based media on the sidelines of a visit to the United Nations.
Earlier on 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 United Arab Emirates-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.