The electronic stopping power (Se) of water vapor (H2O), hydrogen (H2) and oxygen (O2) gases for protons in a broad range of energies, centered in the Bragg peak, was calculated using real-time time-dependent density functional theory (rt-TDDFT) simulations with Gaussian basis sets. This was done for a kinetic energy of incident protons (Ek) ranging from 1.56 keV/amu to 1.6 MeV/amu. Se was calculated as the average over geometrically pre-sampled short ion trajectories. The average Se(Ek) values were found to rapidly converge with 25–30 pre-sampled, 2 nm-long ion trajectories. The rt-TDDFT Se(Ek) curves were compared to experimental and SRIM data, and used to validate the Bragg's Additivity Rule (BAR). Discrepancies were analyzed in terms of basis set effects and omitted nuclear stopping at low energies. At variance with SRIM, we found that BAR is applicable to our rt-TDDFT simulations of 2H2 + O2 → 2H2O without scaling for Ek > 40 keV/amu. The hydrogen and oxygen Core and Bond (CAB) contributions to electronic stopping were calculated and found to be slightly smaller than SRIM values as a result of a red-shift in our rt-TDDFT Se(Ek) curves and a re-distribution of weights due to some bond contributions being neglected in SRIM.