"I've gotta fly" means "I have to leave now." I suppose by extrapolation, "I've got to jet" means the same thing. If the person drives a Jetta, a type of car, it might be a funny way to say "I have to leave now."
Or, did someone ask what type of car he bought? I got a Jetta.
If neither sound right, then please give more context.
Thank you for your reply. I'm sure it has nothing to do with the car. It is a funny film, a comedy, and the scene is this: a couple is breaking up and then she says: Get out. I gotta jetta.
So you are probably right when it means something like: I have to go.