1) In GR gravity is sourced by all types of energy, momentum, stress, pressure, etc. It's not just about mass. So, the Higgs has no special relationship with gravity. It only relates to mass because of the interaction energy that develops between most particles and the vacuum of the Higgs field.
2) The Higgs isn't even responsible for the large majority of the mass we encounter on a daily basis. Most of the mass in ordinary matter is in nucleons - protons and neutrons. When we talk about the composition of nucleons, we talk about them being constructed from up and down quarks; but, the relevant combinations of quarks turn out to only account for a few percent of the nucleon masses. The rest of that mass actually comes from the binding energy that holds the quarks together. This means that most of the mass we actually see is due to the strong force, not the Higgs. And, this mass would be there even if there were no Higgs and the quarks were massless. The more important effect of the Higgs is that it allow the formation of neutral atoms. Were there no Higgs, electrons would be massless and would not be able to be bound into atoms.