The four-body equations of Alt, Grassberger, and Sandhas are solved, for the first time, for proton-3He scattering including the Coulomb interaction between the three protons using the method of screening and renormalization as was done recently for proton-deuteron scattering. Various realistic two-nucleon potentials are used. Large Coulomb effects are seen on all observables. Comparison with data at different energies shows large deviations in the proton analyzing power but quite reasonable agreement in other observables. The effect of the nucleon-nucleon magnetic moment interaction and correlations between p-d and p-3He analyzing powers are studied.