The equilibrium constant for water ionization is the product of activities of H+ and OH- in solution, and each activity is the product of an activity coefficient and a concentration. The activity coefficients (use gamma-+/-) do change as you change the NaCl concentration, because that changes the ionic strength. Since the equilibrium constant depends only on temperature, the concentrations of H+ and OH- must change to balance the changes in gamma-+/-. I am not enough of an expert to point you to a reference on this (beyond p chem text books), but I feel sure it must have been studied.
We found the proton hopping to be faster at low concentrations of solute for another system (see reference). This may hold for NaCl as well.
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The Davies equation is an empirical extension of the Debye-Hueckel limiting law, and it predicts the activity coefficients to be about the same in 0.1 and 1.0 molal solutions. However, it is said to be reliable to about 1.5% in the former but perhaps only ~20% in the latter. Thus, e.g., the actual activity coefficients at 1.0 molal for HCl and NaCl differ by about 25%. I know no easy way to predict any better for H+ and OH- IN 1.0 m NaCl, but it looks like we are dealing with at most 25% changes between 0.1 and 1.0 m NaCl, not factors of 2 or 3 or more. These comments apply, of course, to the equilibrium between water and its ions, with the latter treated in simplest form (H+ and OH-).
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