Starting from a two-dimensional ''hydrogen solid'' model together with the single-electron energy levels of H atom and the two-electron level of H^- ion, we expect single-electron bands originating from the energy levels of the H atoms and a metastable electron-pair band from that of the individual H^- ions. The calculations are performed for periodic potentials of 2D square lattices. We find a metastable electron-pair band above the lowest single-electron band with an energy gap. In order to stabilize the electron-pair states in the crystal, we show the renormalization processes through many-particle exchange-correlation effects within a two-band model, i.e., a higher electron-pair band and a lower single-electron band. Including the many-body effects, we may renormalize the energy bands to find out the conditions in which the electron pairs can become stable. We demonstrate that when the single-electron band is the highest valence band, electron pairs can be stabilized within a wide range of electron-pair and hole densities. It also turns out that when the single-electron band is the lowest conduction band, stabilization of the electron pairs could be expected in more specific crystal potential.