We present a model of thermal X-ray emission from hot spots on the surface of a rotating compact star with an unmagnetized light-element atmosphere. An application to ROSAT, Chandra, and XMM-Newton X-ray observations of the nearest known rotation-powered millisecond pulsar (MSP) PSR J0437--4715 reveals that the thermal emission from this pulsar is fully consistent with such a model, enabling constraints on important properties of the underlying neutron star. We confirm that the observed thermal X-ray pulsations from J0437--4715 are incompatible with blackbody emission and require the presence of an optically thick, light element (most likely hydrogen) atmosphere on the neutron star surface. The morphology of the X-ray pulse profile is consistent with a global dipole configuration of the pulsar magnetic field but suggests an off-center magnetic axis, with a displacement of 0.8-3 km from the stellar center. For an assumed mass of 1.4 M$_{⊙}$, the model restricts the allowed stellar radii to R=6.8-13.8 km (90% confidence) and R>6.7 km (99.9% confidence), which is consistent with standard NS equations of state and rules out an ultracompact star smaller than its photon sphere. Deeper spectroscopic and timing observations of this and other nearby radio MSPs with current and future X-ray facilities (Constellation-X and XEUS) can provide further insight into the fundamental properties of neutron stars. Comment: 10 pages, 8 figures; accepted for publication in the Astrophysical Journal