Since space rocket development is really just military missile development, I guess the next step is SpaceX developing a missile that can be reused.
Yes and no, space launch vehicles are almost all liquid-fueled while almost all modern ballistic missiles are solid-fueled. There are very good reasons for the difference, with space-launch vehicles needing more precise and adjustable burn times while while missiles need the safety, reliability and longevity of solid fuel motors and their a few exceptions like the Minotaur (a converted Peacekeeper ICBM) and the Sarmat...for reasons. So while there is still some stuff that applies to both the cross-politization is less than it used to be when most systems were liquid-fueled.
Isn't the point is a missile to destroy itself? (By crashing into the target or detonating very close?)
If it is a larger missile it may have stages like an SLV and only a warhead would impact the target, but having a reusable missile is not practical for many reasons the least of which would be the landing area would be a great target.