  The measured values of the Standard Model (SM) parameters favors a shallow
metastable electroweak (EW) vacuum surrounded by a deep global AdS or a runaway
Minkowski minimum. Furthermore, fine-tuning is the only explanation for the
Higgs relaxing in its present local minimum. In this paper, assuming no new
physics beyond the SM, we study the universal effect of gravity on the Higgs
dynamics in the early universe. A generic two-parameter model is considered in
which the Higgs is non-minimally coupled to a higher-curvature theory of
gravity. The coupling between the Higgs field and the Weyl field in the
Einstein frame has genuine predictions. In a broad region in the parameter
space, the effective Higgs mass is large and it initially takes over through
fast oscillations. This epoch is followed by the Weyl field slowly rolling a
plateau-like potential. This framework generically predicts that the Higgs
self-coupling in the EW vacuum is enhanced, compared to the SM predictions,
through couplings to the gravity sector. Moreover, when the Higgs is settled in
the EW vacuum, all other scalar flat directions would be lifted via
gravitational effects mediated by the Weyl field.
