  We theoretically investigate the static structure factor of an interacting
Fermi gas near the BEC-BCS crossover at large momenta. Due to short-range
two-body interactions, we predict that the structure factor of unlike spin
correlations $S_{\uparrow\downarrow}(q)$ falls off as $1/q$ in a universal
scaling region with large momentum $\hbar q$ and large scattering length. The
scaling coefficient is determined by the celebrated Tan's contact parameter,
which links the short-range behavior of many-body systems to their universal
thermodynamic properties. By implementing this new Tan relation together with
the random-phase approximation and the virial expansion theory in various
limiting cases, we show how to calculate $S_{\uparrow\downarrow}(q)$ at zero
and finite temperatures for arbitrary interaction strengths, at momentum
transfer higher than the Fermi momentum. Our results provide a way to
experimentally confirm a new Tan relation and to accurately measure the value
of contact parameter.
