The wounded sailor is in a critical condition at a local hospital.
The Fifth Fleet is based in Bahrain to conduct operations across 19.4m sq km of the region that includes the Gulf, Red Sea, Gulf of Oman and parts of the Indian Ocean.
The small Gulf state has been designated as a major non-Nato ally by Washington and also has a free trade agreement with the US.
Around 5,000 US citizens, mostly service personnel, are based in the kingdom.
Earlier this month four Bahrain nationals were charged with with belonging to al-Qaeda after reports that a cell was plotting attacks against US targets in the region.
Three of the suspects are due in court on Tuesday, while a fourth remains at large outside the country.