C arrays that live on the stack have to be created with a constant size (at least before C99). It's a limitation of the language regardless of any machine or stack size.
If we're going to grant that a machine can have infinite ram, surely we can grant that a spec of C can have infinite stack allocated arrays... :)
That's already been granted in this argument. The question now is, given an infinite stack, is this programming style Turing-complete?