We present a precise measurement of the cross section, proton and He-3 analyzing powers, and spin correlation coefficient C-y,C-y for p-He-3 elastic scattering near 65 MeV, and a comparison with rigorous four-nucleon scattering calculations based on realistic nuclear potentials and a model with Delta-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 He-3 binding energy are not successful in reproducing the data. Large sensitivity to the NN potentials and rather small Delta-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-He-3 scattering at intermediate energies is an excellent tool to explore nuclear interactions not accessible by three-nucleon scattering.