Is there a way to use a one-way hash function for both sides to "ratchet" an asymmetric keypair?
Is there any way to derive a series of asymmetric keys, such as using a one-way hashing function, where both sides can predict the next in the sequence?

Here is one way, based on elliptic curves (or finite field based on discrete logs); one such public key encryption method is the Integrated Encyption Scheme.  Now, it does have the drawback that if the attacker learns a prior public key, he can then (with knowledge of the current private key), he can learn the prior private key.  However, this limitation may still be acceptable to you.
In such a scheme, there is a private key (which is an integer $p$ in the range $[0, q)$), and the public key is a point $P = p \cdot G$ (where $\cdot$ is point multiplication if you're using elliptic curves, and exponentiation, more commonly written as $G^p$ if you're using a finite field group).
To ratchet the key forward, the two sides would compute:

Private key: $p' = hash(P) \times p \bmod q$
Public key: $P' = hash(P) \cdot P$

(where $hash$ is a cryptographic hash function).
This works, in that the relationship between the new public key $P'$ and new private key $p'$ still holds; $p' \cdot G = hash(P) \times p \cdot G = hash(P) \cdot P = P'$.
In addition, it is ratcheting; given $p'$, you cannot recover $p$ without knowing the hash of $P$ (and we assumed that he doesn't know that)