There are three (at least) different hull numbers shown for the "Coral Sea". One shot shows the superstructure number with a 65, the hat worn by the CAG has CVN-43 (Correct) on it, finally there is a scene with a life buoy that has Coral Sea CVN-75. The Coral Sea was CVB-43/CVA-43/CV-43 and was conventionally (oil) powered. CV-43 is non-nuclear. CVN-75 is the Harry S. Truman which is a nuclear powered carrier.
The incident with the pilot is supposed to occur aboard the USS Coral Sea some time after the first Gulf War. The Coral Sea was decommissioned in April 1990, and the fighting to free Kuwait began on 16 Jan 1991.