Complexity of determinant of k-minors
Not quite answering what you want, but if you want the gcd of all $k\times k$ minors, then you can compute it in $O(n^\omega)$ or up to logarithmic factors. This is known as Smith Normal Form. For any matrix over principal ideal ring (or PID), the $k$ invariant factor of Smith normal form is the ${\rm gcd}$ of all $k\times k$ minors.