You're right about height, width, depth and time being the standard four dimensions of spacetime. But gravity is not a dimension; it's one of the four fundamental forces, the other three being electromagnetism, the strong nuclear force and the weak nuclear force.
however, observation has shown that the structure of space time is bent into "gravity wells" around massively large objects. Here are two sentences. 1.)Space tells matter how to move. 2.)Matter tells space how to bend. Sentence one describes "height, width, depth and time being the standard four dimensions of spacetime." Sentence one describes how we understand reality. However, all four dimensions seem to be subject to something else described in sentence two. The bend or gravity. But why is this bending of space described as a force rather than the character of space itself. It doesn't seem as if space is empty. Rather it seems that if space can bend than it is more like a material.
Either that or our notion of dimensions is a function of newtonian mathmatics. We're biased or our point of view is in favor of space time rather than matter time--if there is such a beast. I'm no mathmatician but I've heard recently some speculation at free republic that this bias had something to do with Newton's work and that it could be rectified or rebalanced by Newton's contemporary, Liebnitz, who did some work with infintessimals, a number system that is built around a one dimensional zero--or something like that. At this point I'm out of my depth.
Rather it seems that if space can bend than it is more like a material.
The basic equation in Einstein's General Relativity places (essentially) the geometric structure of spacetime on one side and stress-energy on the other, that is, there is an equivalence between the mathematical (geometry) and the physical (stress-energy). So, yes, spacetime (not just space) is more like a material, at least in General Relativity.
John Baez has a nice tutorial on General Relativity, but it's not for the faint of heart: