There has been a recent trend to reduce the size of photonic waveguide devices to enable high-density integration in silicon photonic integrated circuits. However, this miniaturization tends to result in increased polarization dependency. Particularly challenging is designing devices based on ring waveguides with small radii, which exacerbates the polarization sensitivity. For these microring resonators, a legitimate question is then: Is it possible to simultaneously maintain the conditions of single-mode and structural polarization independence while shrinking the size of both the bend radius and the waveguide cross section, and, if so, how small can the ring resonator be? We demonstrate theoretically the feasibility of achieving this via deeply etched submicrometer silicon-on-insulator rib waveguides, and we show that, for a given cladding and core thickness, the radius of a polarization independent microring resonator can be as small as 3 microm, being limited chiefly by the residual birefringence of the resonator cavity and the bend losses.