"Homomorphic" one way function for public/private key pairs
Sure. ​ Let (Gen,Sign,Ver) be any signature scheme, and let h be any pseudorandom generator from {0,1}k to {0,1}k+L. ​ (Since h is a pseudorandom generator, h is one-way.)

Gen' takes k+L random bits as input and outputs [those bits as the private key] and

[the public key resulting from running Gen on the last L of those bits] as the public key.

Sign' outputs the result of running Sign on [the message and

[the private key resulting from running Gen on the last L bits of the (k+L)-bit private key]].

(Gen',Sign',Ver) is a signature scheme such that "we can efficiently calculate

the public key associated with $h(s_k)$" without even needing $s_k$'s public key.

That makes me think you're actually after something stronger.