We have extended the tunneling ionization model of Ammosov-Delone-Krainov (ADK) for atoms to diatomic molecules by considering the symmetry property and the asymptotic behavior of the molecular electronic wave function. The structure parameters of several molecules needed for calculating the ionization rates using this molecular ADK model have been obtained. The theory is applied to calculate the ratios of ionization signals for diatomic molecules with their companion atoms that have nearly identical binding energies. The origin of ionization suppression for some molecules has been identified. The predicted ratios for pairs with suppression $({\mathrm{D}}_2:\mathrm{Ar},$ ${\mathrm{O}}_2:\mathrm{Xe})$ and pairs without suppression $({\mathrm{N}}_2:\mathrm{Ar},$ CO:Kr) are in good agreement with the measurements. However, the theory predicts suppression for ${\mathrm{F}}_2:\mathrm{Ar},$ which is in disagreement with the experiment. The ionization signals of NO, ${\mathrm{S}}_2,$ and of SO have also been derived from the experimental data, and the results are also shown to be in agreement with the prediction of the present molecular ADK theory.