  To reveal the origin of the "nematic electronic fluid phase" in
Sr$_3$Ru$_2$O$_7$, we apply the self-consistent vertex correction analysis to
the ($d_{xz},d_{yz}$)-orbital Hubbard model. It is found that the
Aslamazov-Larkin type vertex correction causes the strong coupling between spin
and orbital fluctuations, which corresponds to the Kugel-Khomskii spin-orbital
coupling in the local picture. Due to this mechanism, orbital nematic order
with $C_2$ symmetry is induced by the magnetic quantum criticality in
multiorbital systems, while this mechanism is ignored in the
random-phase-approximation. The present study naturally explains the intimate
relation between the magnetic quantum criticality and the nematic state in
Sr$_3$Ru$_2$O$_7$ and Fe-based superconductors.
