To study the molecular damage induced in the form of single-strand and double-strand breaks by ionizing radiation at the DNA level, the Geant4-DNA Monte Carlo simulation code for complete transportation of primary protons and other secondary particles in liquid water has been employed in this work. To this aim, a B-DNA model and a thorough classification of the complexity of the DNA damage were used. Strand breaks were assumed to have primarily originated by direct physical interactions via energy depositions, assuming a threshold energy of 17.5 eV, or indirect chemical reactions of hydroxyl radicals, assuming a probability of 0.13. The simulation results on the complexity and frequency of various damages are computed for proton energies of 0.5-20 MeV. The yield results for a cell (Gy cell)-1 are presented, assuming 22 chromosomes per cell and a mean number of 245 Mbp per chromosome. The results show that for proton energies below 2 MeV, more than 50% of the energy depositions within the DNA volume resulted in strand breaks. For double-strand breaks (DSBs), there is considerable sensitivity of DSB frequency to the proton energy. A comparison of DSB frequencies predicted by different simulations and experiments is presented as a function of proton linear energy transfer (LET). We show that our yield results (Gy Gbp)-1 are generally comparable with various experimental data and there seems to be a better agreement between our results and a number of experimental studies when compared to other simulations.