Even if √2 is rational, it is not an integer, and (especially in your second paragraph) you are trying to do things with it that only make sense with integers.

I don't think that continuing this conversation can be productive, since you seem to be objecting to standard techniques. A rigorous textbook in number theory can probably explain the proof that √2 is irrational more thoroughly, and you can follow back the lemmas and see exactly where your confusion lies.