No, you cannot recover the message hash from an ECDSA signature. The siagnture is calculated by generating
(x, y) = kG where
k is the secret nonce and
G is the generator for the curve. Then
r = x and
s = k^(-1)(z + rd) where
z is the message and
d is the private key. We can retrieve the public key
dG by seeing that
r^(-1) (kGs - zG) = dG. But trying to do the same for
z will only ever result in
zG, since we do not know
zG is exactly the DLP on the elliptic curve and is not feasible to compute.
Edit 2: It should be noted that while computing the message hash is infeasible, it is possible to check guesses, if the message was not salted before hashing (which it usually isn't). So if the message comes from a small set of possibilities then it can be recovered.
Edit: links for further reading