We present a precise measurement of the cross section, proton and He3 analyzing powers, and spin correlation coefficient Cy,y for p-He3 elastic scattering near 65 MeV, and a comparison with rigorous four-nucleon scattering calculations based on realistic nuclear potentials and a model with Δ-isobar excitation. Clear discrepancies are seen in some of the measured observables in the regime around the cross section minimum. Theoretical predictions using scaling relations between the calculated cross section and the He3 binding energy are not successful in reproducing the data. Large sensitivity to the NN potentials and rather small Δ-isobar effects in the calculated cross section are noticed as different features from those in the deuteron-proton elastic scattering. The results obtained above indicate that p-He3 scattering at intermediate energies is an excellent tool to explore nuclear interactions not accessible by three-nucleon scattering.