At one point the drone was just 30m (100ft) from the plane which had to take evasive action to avoid a collision, officials said.
A navy spokesman accused the Iranians of behaving in an "unsafe and unprofessional" manner.
Tensions between the countries are high with the Gulf a potential flashpoint.
US Naval Forces Central Command spokesman Commander Bill Urban said the incident happened in international air space as an F/A-18E Super Hornet was preparing to land on the USS Nimitz.
Despite radio warnings to stay clear of the carrier, the drone carried out "unsafe and unprofessional" altitude changes near the jet, he said.
"The dangerous manoeuvre by the QOM-1 [drone] in the known vicinity of fixed-wing flight operations and at coincident altitude with operating aircraft created a collision hazard and is not in keeping with international maritime customs and laws," Cdr Urban said.
The incident is the latest in a series of close encounters in the Gulf between US and Iranian forces.
In July, a US Navy ship fired warning shots at an Iranian Revolutionary Guard vessel after it came within 140m (450ft) and ignored radio calls.
In January, a US destroyer fired three warning shots when Iranian fast-attack vessels approached at high speed in the Strait of Hormuz.
An Iranian drone has flown dangerously close to a US fighter jet as it was about to land on a carrier in the Gulf, the US Navy says.