Again, wrong. It is not "elemental". Dissolved oxygen is molecular oxygen, O2. That said, it is still an oxidizer. You will not find elemental oxygen around in nature, as it will react with another oxygen and from ozone. Look at electrolysis, you've just broken apart a water molecule, and the "elemental" oxygen rapidly forms diatomic, molecular oxygen.
(At very high temperatures, like >5000K, you can get dissociation of molecular oxygen, but not at any temps we are interested in.)
When air dissolves in water, the oxygen does not ionize. It simply dissolves. Same with nitrogen. They are both diatomic molecules and they do not ionize in water.
Think about it, guys. If oxygen did "decompose" and raise the pH of water, all of our oceans, rivers, and other natural bodies of water would eventually become caustic and nothing could live in them...