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.

DUP leader Arlene Foster said the "wide-ranging" pact was "good for Northern Ireland and the UK" -  so where will the money be spent?
The heart of the financial package is infrastructure spending - £400m of it over two years.
That is a significant sum given that Northern Ireland's annual infrastructure budget is about £1bn.
The deal specifically mentions the York Street Interchange, a project to ease congestion where three major roads meet in Belfast.
It is expected to cost about £150m, so there should be plenty left over for other major projects.
They could include a planned transport hub in the centre of Belfast and other major roads schemes like the completion of the A5 or A6.
On top of the £400m, there is a further £150m for ultra-fast broadband.
This was a project which was already being scoped by the Department of the Economy before the collapse of Stormont.
It is ultimately about improving the connectivity of rural businesses.
Previous Stormont/Westminster deals have also included infrastructure but they typically involved allowing Stormont to borrow more.
This time it is cold, hard, no-strings-attached cash.
The money for health and education is more modest - given that a figure of £1bn for health alone was being mentioned during the negotiations.
There is a minimum of £250m for health and and we can assume it will get about half the "immediate pressures" money, bringing it to a total of £300m.
That will help ease the problems with waiting lists and give the system some breathing space ahead of a major reform programme.
The "immediate pressures" money for education (about £50m) will go some way to easing a growing row with head teachers who have said they will not impose any more cuts.
There is also £100m to be spent on deprived communities over five years.
It will be interesting to see what the delivery mechanism for that money will be.
Stormont's Social Investment Fund, which was also designed to help those communities, has been mired in controversy.
Another part of this deal is to give flexibility to a previously agreed £500m fund for investment in shared education and housing projects.
Stormont has struggled to spend this money, in its first year just £3m of a £50m allocation was spent.
So its is possible that the new flexibility will allow all that money to go into general infrastructure spending.
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.
[NEXT_CONCEPT]
Northern Ireland will receive an extra £1bn over the next two years as part of the deal that will see the Democratic Unionist Party's 10 MPs back Theresa May's minority government in Commons votes.