Fist of all, saying that there is one hydrogen ion for $10^6$ ions of water is wrong.ph is the negative of log of concentration of hydrogen ions. pH 6 means that there is $10^{-6}$ moles of hydrogen in a litre of the solution. Moreover, in a battery the electrolyte gets depleted not the electrodes. So, the voltage will be constant till the electrolyte can provide the potential ideally.

In reality, the voltage curve drops at the end of the life of the battery.