  We study gravitational interaction of Higgs boson through the unique
dimension-4 operator $\xi H^\dag H R$, with $H$ the Higgs doublet and $R$ the
Ricci scalar curvature. We analyze the effect of this dimensionless nonminimal
coupling $\xi$ on weak gauge boson scattering in both Jordan and Einstein
frames. We explicitly establish the longitudinal-Goldstone boson equivalence
theorem with nonzero $\xi$ coupling in both frames, and analyze the unitarity
constraints. We study the $\xi$-induced weak boson scattering cross sections at
O(1-30)TeV scales, and propose to probe the Higgs-gravity coupling via weak
boson scattering experiments at the LHC(14TeV) and the next generation pp
colliders (50-100TeV). We further extend our study to Higgs inflation, and
quantitatively derive the perturbative unitarity bounds via coupled channel
analysis, under large field background at the inflation scale. We analyze the
unitarity constraints on the parameter space in both the conventional Higgs
inflation and the improved models in light of the recent BICEP2 data.
