  Nematic order resulting from the partial melting of density-waves has been
proposed as the mechanism to explain nematicity in iron-based superconductors.
An outstanding question, however, is whether the microscopic electronic model
for these systems -- the multi-orbital Hubbard model -- displays such an
ordered state as its leading instability. In contrast to usual electronic
instabilities, such as magnetic and charge order, this fluctuation-driven
phenomenon cannot be captured by the standard RPA method. Here, by including
fluctuations beyond RPA in the multi-orbital Hubbard model, we derive its
nematic susceptibility and contrast it with its ferro-orbital order
susceptibility, showing that its leading instability is the spin-driven nematic
phase. Our results also demonstrate the primary role played by the $d_{xy}$
orbital in driving the nematic transition, and reveal that high-energy magnetic
fluctuations are essential to stabilize nematic order in the absence of
magnetic order.
