We present the first two-dimensional (2-D) Alfv¿n wave turbulence--driven solar wind model which takes the proton temperature anisotropy into account. While the modeled proton temperature anisotropy in the fast solar wind is established in the inner corona and yields Tp$parallel$/Tp$perp$ = 0.57 at 1 AU, which is comparable to measured values, Tp$parallel$ and Tp$perp$ are only about half the observed values. In the slow wind, on the other hand, the modeled values for Tp$parallel$ and Tp$perp$ as well as their ratio are close to those measured in interplanetary space. Curiously, the dip in the velocity that develops near the cusp at the top of the helmet streamer reduces the effect of transverse expansion and leads to a realistic electron temperature in the slow wind at 1 AU, although no explicit external heating is applied to electrons. Comparison with models with and without proton temperature anisotropy shows that by allowing the proton temperature anisotropy to develop, the average proton temperature is lower than the isotropic case primarily because of the cooling in the direction parallel to the magnetic field. These results imply that ion cyclotron resonance models with isotropic proton temperature are somewhat optimistic in assessing the role of Alfv¿n wave turbulence in driving the fast solar wind. Inclusion of the temperature anisotropy of protons and proton thermal conduction are necessary for any physically realistic model.