<p>We present a theory for determining the linear dimensions of compact rectangular microwave patch antennas on metamaterial substrates with a high real part of the effective relative permittivity. This theory demonstrates that significant miniaturization of the volume profile of such antennas is achievable with enhanced performance using a metamaterial substrate instead of a dielectric substrate. It is assumed that the metamaterial substrate is a host dielectric medium with periodically embedded metallic inclusions. The proposed theory is based on a simple analytical algorithm design to minimize the volume profile of the antenna patch. It establishes a relationship between the effective relative permittivity of the substrate, the resonant frequency, and the substrate thickness. The proposed approach achieves up to 80% reduction in the antenna volume profile. Notably, the proposed optimization approach does not impose any restrictions on the geometry of the metamaterial unit cell used to create the antenna substrate except for the case of positive values of the effective relative permittivity and permeability. Furthermore, it does not require substantial computational resources for designing the linear dimensions of patch antennas. The derived relations are intended to be used along with modern electromagnetic simulators for the CAD design of compact microwave metamaterial patch antennas with a rectangular patch and the substrate with cylindrical copper inclusions of circular cross section. The proposed optimization theory is validated through an electromagnetic simulator based on the finite difference time-domain method. Moreover, appropriate computer simulations have shown that employing metamaterials in place of conventional dielectric materials to create the substrate not only leads to the miniaturization of the antenna but also enhances its overall performance.</p>