The vector analyzing power A(y)(theta) of the H-2(p over arrow pointing right,p)H-2 scattering has been measured at E(p) = 5, 6, 6.5, 7, 8, 8.5, 9, 10, 12, 14, 16, and 18 MeV with a typical statistical accuracy of 0.0009 and an uncertainty in the beam polarization of less than 0.7%. The differential cross section sigma(theta) of the H-2(p,p)H-2 scattering has also been measured with a typical uncertainty of 0.8% using unpolarized beams of the same energies and of 2, 2.5, 3, and 4 MeV. A large discrepancy of about 25% between the experiment and the Faddeev calculation with the Paris NN potential is observed at the maximum of A(y)(theta) around theta(c.m.) = 120-degrees. This discrepancy is shown to be reduced by modifying LS force in the NN potential, though a discrepancy still remains in the energy dependence of the A(y)(theta) maximum. At the minimum of sigma(theta) around theta(c.m.) = 120-degrees, a large discrepancy ranging from -19% at 2 MeV to +24% at 18 MeV is observed between the experiment and the calculations with either of the original and the LS-modified NN potentials. This indicates that an improvement is also necessary in the scalar part of the potential. The difference in the A(y)(theta) maximum between the p over arrow pointing right + d and the n over arrow pointing right + d scatterings is discussed.