The dead man and the three injured people are reported to be Indian fishermen.
It was not immediately clear why the USNS Rappahannock attacked a small motor boat near the Dubai port of Jebel Ali on Monday.
US consular officials in Abu Dhabi have confirmed the incident but refused to provide further details.
Local officials say the boat could have been mistaken as a threat.
The US Navy’s 5th Fleet, which is based in Bahrain, says it is investigating the incident.
Washington recently expanded its military presence in the Persian Gulf, sending an unspecified number of F-22 stealth fighters and warships to the region.