Protons and osmotic potential

If the water is pure on both sides, it will have the same pH (assuming the same temperature and pressure). Thus the only way to create a difference in pH would be through the presence of conjugate acids and/or bases other than $ce{H_2O}$, $ce{H^+}$ and $ce{OH^-}$.

And since you've specified that your membrane is permeable only to water, these conjugate acids and bases would necessarily act as osmolytes. I.e., their presence would lower the chemical potential of the water, causing the water to move towards the direction with the higher osmolyte activity. So whether the water moves one direction or the other depends on the relative activities of the conjugate acids and/or bases you have used to create the pH difference.

No, because osmosis is not an electrical phenomenon, but rather a chemical one. The concentration of protons would diffuse across the membrane if the fluid, I'm assuming which is water, will be allowed to interact with the other side of the membrane. However, water will not seek to increase the pH of an acidic solution within a membrane the same way it will to dilute concentrations of ions. The resulting balance of pH across the membrane will be as a result of the water to water interaction across the membrane and not because of the pH being imbalanced.