The self-consistent Ornstein-Zernike approach (SCOZA) is solved numerically for a lattice gas or Ising model on the simple square lattice in two dimensions. Interactions of varying range are considered, and the results are compared with corresponding simulation ones. We focus especially upon the location of the critical temperature T-c which is identified with the maximum of the specific heat. The maximum remains finite for the finite-sized simulation sample and also for SCOZA, which treats infinite lattices in two dimensions as though they were finite samples. We also investigate the influence of the precise form of the interaction, first using an interaction that extends the nearest-neighbor case in a simple way and then considering the square-well interactions used in the simulations. We find that the shift in T-c away from its mean-field value is governed primarily by the range of interaction. Other specific features of the interaction leave a smaller influence but are relevant to a quantitative comparison with simulations. The SCOZA yields accurate results, and the influence of the precise form of the attractive interaction plays a significant role in SCOZA's success.