Show that limit does not exist (two variables)

What you have done is correct. The limit exists only if the value of the limit along every direction that leads to $(0,0)$ is same.

So when you calculate $$\lim_{x\to 0}\frac{x^2y^2}{x^2y^2+(x-y)^2}$$ you are calculating limit along the line $x=0$.

Similarly,
$$\lim_{y\to 0}\frac{x^2y^2}{x^2y^2+(x-y)^2}$$ is limit along line $y=0$.

And the last limit you calculated is along line $y=x$.

So to answer your question, yes it would have been perfectly acceptable if you did not calculate limit along $y=0$. Just showing two examples where the limit comes out to be different along different directions is enough to show limit does not exist.

Your reasoning is ok. If the limit existed, you would obtain the same value for every directional limit, which was not the case.