Hi gris, it due to the gas by the name of propanethiol S-oxide released by the onion when it is peeled. This gas is a rich sulfur based compound as you may have guessed by the name. When this gas is released, it induces a chemical reaction with the water which is being produced constantly to keep the eyes moist. The reaction causes sulphuric acid to be formed and irritation starts. The glands then shed more water to clear the eyes of the irritating substance.

This is the reason why cutting onion in water doesn't induce a reaction because the gas mixes with water and cannot reach the eyes.