Can a civilian purchase a new (Not Retired) fighter jet? Let's take a jet like the F22 for example. Strip all the classified systems/weaponry out of it (civilian wouldn't need that), and just leave the basic aircraft itself. 

Could a civilian technically purchase this aircraft for personal recreational use/travel? Or are these exclusively contracted to the military? How does that work legally/technically? I've always been curious.

There are multiple cases of wealthy civilians purchasing USED fighter jets, sure, but I'm talking about a wealthy civilian who can afford it ordering a shiny new (not retired, there is a difference) personalized, non-armed F22 for personal use. Is that even a legitimate premise?
Certain jets, yes.

But not the F22.

See current listings:

https://www.controller.com/listings/aircraft/for-sale/list/category/10072/turbine-military-aircraft