The microscopic polymer reference interaction site model theory of polymer nanocomposites composed of flexible chains and spherical nanoparticles has been employed to study second virial coefficients and spinodal demixing over a wide range of interfacial chemistry, chain length, and particle size conditions. For hard fillers, two distinct phase separation behaviors, separated by a miscibility window, are generically predicted. One demixing curve occurs at relatively low monomer-particle attraction strength and corresponds to a very abrupt transition from an entropic depletion attraction-induced phase separated state to an enthalpically stabilized miscible fluid. The homogeneous mixture arises via a steric stabilization mechanism associated with the formation of thin, thermodynamically stable bound polymer layers around fillers. The second demixing transition occurs at relatively high monomer-particle adsorption energy and is inferred to involve the formation of an equilibrium physical network phase with local bridging of particles by polymers. This spinodal is sensitive to both particle-monomer diameter ratio and the spatial range of the interfacial attraction. The miscibility window narrows, and can ultimately disappear, with increasing polymer chain length, direct van der Waals attractions between fillers, and/or particle-monomer size asymmetry ratio. The implications of our results for the design of well-dispersed thermodynamically stable polymer nanocomposites, and the formation of nonequilibrium gels, are discussed.