  We consider the standard Einstein-Hilbert-Higgs action where the Higgs field
couples nonminimally with gravity via the term $\xi h^2 \mathcal{R}$, and
investigate the stability of the electroweak vacuum in the presence of
gravitational corrections in both the metric and Palatini formulations of
gravity. In order to identify the differences between the two formalisms
analytically, we follow a perturbative approach in which the gravitational
corrections are taken into consideration via a leading order expansion in the
gravitational coupling constant. Our analysis shows that in the Palatini
formalism, the well-known effect of gravity suppressing the vacuum decay
probability, becomes milder in comparison with the metric case for any value of
the nonminimal coupling $\xi$. Furthermore, we have found that in the Palatini
formalism, the positivity of the gravitational corrections, which is a
necessary requirement for the unitarity of the theory, entails the lower bound
$\xi>-1/12$.
