I'm not sure what you're quite asking about the second reaction. I believe you may be referring to it not running to completion. This is because reactions tend to favor the direction of the weak acid and base. Here H20 definitely functions as an acid and has a certain pka value. Remember, the lower the pka value the strong the acid. However, the pka value of nitric acid is much lower than that of H20, I believe its negative, which means its very strong. The reaction is going to favor the left side. Similarly, OH- is a strong base than NO3- which is very weak, since its acid, Nitric acid is very strong. Remember, acid-conjugate base strength is inversely proportional.
HNO3 + OH- --> NO3- + H20 is more likely to occur.