Non-Equilibrium Molecular Dynamics simulations were performed to examine in detail the mechanisms involved in a reverse osmosis process for the removal of monovalent (Na+, Cl−) and divalent (Mg2+) ions from saline water, using multilayered glycine-functionalized graphene oxide (GO) membranes. We varied parameters such as the degree of functionalization of the GO flakes and the structural flexibility of the membranes and examined their performance in a wide range of applied pressure gradients. In all models examined, water permeability was found to be almost 2 orders of magnitude larger compared to that in conventional reverse osmosis (RO) membranes. High structural flexibility of the membranes was found to increase water permeability due to the formation of additional nanochannels that favor faster water transport. Functionalization of the GO flakes by the glycinate groups resulted in a decrease in water permeability due to the formation of a denser hydrogen-bonding network within the membranes. Three main mechanisms were identified regarding ion selectivity. Due to the overall negative charge of the membranes, the strong adsorption of the Mg2+ ions onto the GO flakes resulted in their full rejection. Weaker adsorption of the Na+ ions led only to their partial rejection. At the same time, the electrostatic repulsion of the Cl− ions by the membranes was found to result in almost full removal levels at high pressure gradients. The rejection of the monovalent ions was found to decrease with the structural flexibility of the membranes and increase with the functionalization of the GO flakes, in response to the changes observed in water permeability on these membrane modifications. The pressure-dependent deformability of the structurally flexible membranes combined with charge accumulation close to the membrane's entrance, resulted in a non-monotonic dependence of the monovalent ion rejection on the applied pressure difference.