The USS Boxer, "took defensive action" against the Iranian aircraft on Thursday as it was "threatening the safety of the ship and the ship's crew", President Donald Trump announced at the White House.
"The drone was immediately destroyed," he added without giving more details.
Iranian Foreign Minister Mohammad Javad Zarif said, "We have no information about losing a drone".
The Pentagon said the incident happened at 10am local time (07:00 GMT) on Thursday in international waters while the USS Boxer was transiting the waterway to enter the Gulf.
"A fixed-wing unmanned aerial system (UAS) approached Boxer and closed within a threatening range," chief Pentagon spokesman Jonathan Hoffman said in a statement. "The ship took defensive action against the UAS to ensure the safety of the ship and its crew."
The downing of the drone comes at a time of mounting tensions in the Gulf 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 facilities, and interests and calls upon all nations to condemn Iran's 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."
No immediate response was issued by Iran.
Iran shot down a US drone last month that it said was flying over its territory and the escalation raised concerns the two countries might go to war.
Trump ordered a retaliatory military attack but called it off at the last moment, saying the risk of casualties was disproportionate to the shootdown by Iran, which did not cost any US lives.
Tensions have soared since Trump unilaterally abandoned a landmark 2015 nuclear deal with world powers aimed at curbing Iran's nuclear programme last year, hitting Tehran with crippling sanctions that it calls "economic terrorism".
The US has blamed Iran for a series of attacks since mid-May on shipping around the Strait of Hormuz, the world's most important oil artery - accusations Tehran rejects.
Earlier on Thursday, Washington demandedIran immediately release a vessel it seized in the Gulf, and a US military commander in the region said the US would work "aggressively" to ensure free passage of vessels through the vital waterway.
Iran's Revolutionary Guard seized the foreign tanker accused of smuggling oil with a crew of 12 on Sunday, Iranian state TV reported.