Our immune system is trained how to differentiate between what is supposed to be there and what isn’t only on things floating around in the blood. Certain parts of the human body - like the liquids inside our eyes - are permanently isolated from the bloodstream, and so our immune system will never have encountered anything like it. That’s why if it ever enters the bloodstream it can trigger an immune response.

If our eyes can absorb oxygen from tears then they can certainly do so when submerged provided the water is sufficiently oxygenated.