So, the Higgs is, if I recall right, sensitive to the masses of other particles, and I don't think this has much to do with gravity per say (gravity just reacts to mass-energy to curve spacetime) but the fact that Higgs can decay to other particles and also feels the 'Higgs' effect/field of which the Higgs boson is kind of like a left over. The Higgs mass can thus 'blow up' from contributions from other particles, because in a quantum field a particle will potentially fluctuate to several particles and then back again, and of course, you can't decay into something if you are less energetic than it.

I very hazily recall that it is possible to have some mass from non-Higgs effect sources, for instance quarks binding to each other contributes most of a proton's mass rather than the Higgs effect, so the Higgs boson could have /some/ mass even if we turned off all the Higgs bits except for the boson itself, but my impression was that it was majority from Higgs interactions, from which the Higgs boson is relatively 'unprotected' from being ballooned up. A counterpart particle can provide counterterms to help keep mass low, like a seesaw, but the standard model Higgs has no such counterpart unless you introduce something extra.