Discrete logarithm with exponentiation oracle
No, there is no known way to recover $d$ using an Oracle that, given $m$, returns $m^d \bmod n$; there is also no proof that such a method does not exist.

Such a method would imply that the RSA problem (which is, given $m^e \bmod n$, recover $m$) is equivalent to the factorization problem (which knowledge of $d, e$ would allow you to solve); it is unknown if these two problems are equivalent.