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We consider collisional excitation of H₂ molecules in C-type shocks propagating in dense molecular clouds. New data on collisional rate coefficients for (de-)excitation of H₂ molecule in collisions with H atoms and new H₂ dissociation rates are used. The new H₂-H collisional data are state of the art and are based on the most accurate H₃ potential energy surface. We re-examine the excitation of rotational levels of H₂ molecule, the para-to-ortho-H₂ conversion, and H₂ dissociation by H₂-H collisions. At cosmic ray ionization rates ζ >= 10^–16 s^–1 and at moderate shock speeds, the H/H₂ ratio at the shock front is mainly determined by the cosmic ray ionization rate. The H₂-H collisions play the main role in the para-to-ortho-H₂ conversion and, at ζ >= 10^–15 s^–1, in the excitation of vibrationally excited states of H₂ molecule in the shock. The H₂ortho-to-para ratio of the shocked gas and column densities of rotational levels of vibrationally excited states of H₂ are found to depend strongly on the cosmic ray ionization rate. We discuss the applicability of the presented results to interpretation of observations of H₂ emission in supernova remnants.