Since the proposed Higgs field is a quantum fluid filling all of space, what is it's relationship to gravity?
Is gravity the result of pressure created by the Higgs Quantum Fluid?

Nope, it's something different altogether. Now, the Higgs field has an effect on the behavior gravity, in that interactions with the Higgs field are what give particles their masses. But gravity isn't "created" by the Higgs field: it would be around regardless.

Gravity, so far as we know, is the way that matter warps space-time, combined with how matter moves through warped space-time. The quantum-mechanical underpinnings of precisely how this operates are as yet uncertain, but some things are clear: gravity must be mediated by a massless spin-2 field. It must be massless (or nearly so) because gravity is a long-range force. It must be even-integer-spin because it is always attractive. A force based upon a spin-0 field would cause a different deflection of photons, so that's out. A force based upon a spin-2 massless field, however, replicates Einstein's equations (in some appropriate classical limit).

Well, we don't know. But then, why are you singling out gravity here? This question can be asked of any of the four forces we know.

It seems so. We have found no deviation so far.

The boson which is the quantum of the field is a spin-2 particle. If you don't know what that means, read up a bit on quantum-mechanical spin. Wikipedia is, as usual, a decent place to start (link here).