Spectrum restrictions in the signature consisting of just a single binary operation In the signature {*}, where * is an operator of arity 2, is there a theory whose spectrum is the set of prime powers?
Finite models of field theory have the cardinality of prime powers. Of course, they don't quite fit your signature.

However, the answer to this question demonstrates methods to capture both addition and multiplication in a single binary operation. For each constant and function defined, you will have to 'reverse skolemize' into a FOL sentence, 'and' them all together with unique variable names, and then 'and' all the axioms that use these functions into one sentence (this ensures the 'reverse skolemized' function is the same for each axiom).

Actually performing all this would be a nasty bit of work, but I believe this information is sufficient to demonstrate the answer to your question is an affirmative.
Add to group theory an axiom asserting that every element's order is a power of some prime $p$.

Theorem 1.1 in this paper:
http://www.math.uconn.edu/~kconrad/blurbs/grouptheory/cauchyapp.pdf