RSA Without Padding?
No, an attack against RSA without padding that let one recover messages does not, as far as we know, degenerate into leaking a working private exponent $d$, or factoring $n$. In fact, even unlimited access to a decryption oracle performing $c\mapsto m=c^d\bmod n$ for any $c$ will not allow that either.

To back up the above, see this survey by Dan Boneh where it is Open Problem 1.