The relationship between the Higgs field and gravity is not as direct as you described. Let's break down the concepts involved.
The Higgs field is a fundamental field in particle physics that permeates all of space. Particles interact with the Higgs field, and through this interaction, they acquire mass. The Higgs field is responsible for the mechanism by which elementary particles, such as quarks and electrons, acquire their masses.
On the other hand, gravity is described by the theory of general relativity, which explains the force of gravity as the curvature of spacetime caused by mass and energy. In general relativity, the presence of mass and energy warps the fabric of spacetime around it, causing objects to move along curved paths.
The curvature of spacetime, and hence the effect of gravity, is not directly related to the Higgs field. The Higgs field primarily influences the masses of particles, while gravity is a result of the curvature of spacetime caused by the distribution of mass and energy.
It's worth noting that there is ongoing research and exploration into the unification of general relativity and quantum mechanics, which includes understanding the role of the Higgs field in the context of gravity. However, at the present state of scientific understanding, the direct connection you mentioned between the gradient of the Higgs field and the effect of gravity is not supported by current theories.