I think a counterexample can be obtained via simplicial spheres. It is known that in higher dimensions, most simplicial spheres do not belong to a polytope. The dual of the face lattice of such a $d$-dimensional non-polytopal simplicial sphere is a $d$-regular "simple" poset in your sense, and cannot belong to a $d$-polytope (as otherwise, the original poset would too).
Note
While such a poset does not belong to a $d$-polytope, it is not immediately clear that its graph does neither. If I understood correctly, in David's answer he constructs an example where neither the poset (a CW complex) nor its graph belong to a polytope.