In a hypothetical enclosed membrane only permeable to water (including hydronium), where inside the pH is lower than the outside, would water move inside?
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.