Thanks for your replies.
Wow! This gets way beyond me..."spin" is just "bolted on" to the "Hilbert space"...! Don't have a clue what that's all about.
Anyway, the consensus seems to be--
No, relativity is not logically necessary, at least as far as anyone can tell.
I guess I was thinking that if you can think through the implications of relativity in this universe, then you could also do the same for the opposite case--the case of a universe that is exactly the same as this one--except--without the principle of relativity.
But I guess you can't do that, it would be too difficult.
I gather that what you can say is just that without relativity, the universe would be very different from how it is.