We study the postinflationary dynamics of the Standard Model (SM) Higgs field in the presence of a nonminimal coupling ξ|Φ|2R to gravity, both with and without the electroweak gauge fields coupled to the Higgs field. We assume a minimal scenario in which inflation and reheating are caused by chaotic inflation with a quadratic potential, and no additional new physics is relevant below the Planck scale. By using classical real-time lattice simulations with a renormalization group improved effective Higgs potential and by demanding the stability of the Higgs vacuum after inflation, we obtain upper bounds for ξ, taking into account the experimental uncertainty of the top-Yukawa coupling. We compare the bounds in the absence and presence of the electroweak gauge bosons and conclude that the addition of gauge interactions has a rather minimal impact. In the unstable cases, we parametrize the time when such instability develops. For a top-quark mass mt≈173.3  GeV, the Higgs vacuum instability is triggered for ξ≳4–5, although a slightly lower mass of mt≈172.1  GeV pushes up this limit to ξ≳11–12. This, together with the estimation ξ≳0.06 for stability during inflation, provides tight constraints to the Higgs field-curvature coupling within the SM.