We study the charmonium spectrum using a complete one gluon exchange approach based on a phenomenological relativistic $q\bar{q}$ potential model with Dirac spinors in momentum space. We use phenomenological screening factors to include vacuum quantum effects. Our formulation does not rely on nonrelativistic approximations. We fit the lowest-lying charmonia (below the $D\bar{D}$ threshold) and predict the higher-lying resonances of the spectrum. In general, we reproduce the overall structure of the charmonium spectrum and, in particular, we can reasonably describe the $X(3872)$ resonance mass as (mostly) a $c\bar{c}$ state. The numerical values of the free parameters of the model are determined taking into account also the experimental uncertainties of the resonance energies. In this way, we are able to obtain the uncertainties of the theoretical resonance masses and the correlation among the free parameters of the model.