You are right, from what I have read about the F-4, no way he could have brought that off, a much trickier and more complex machine with deadly quirks.
"On July 4, 1986, 21-year-old Marine Lance Corporal, Howard Foote Jr., an aviation mechanic at El Toro, took an A4 Skyhawk on an unauthorized 90 minute joyride over southern California. Foote, an accomplished glider pilot, was despondent after recently learning that due a medical condition he would never be able to fly in the Marines".
According to subsequent news reports, LCpl Foote was not court-martialed but got an Other-Than-Honorable discharge as his punishment.