Remove the apostrophe from 'doesn't'.

“Just because I was mean to you, it does not mean you should be mean to me.” OR
“Just because I was mean to you, does not mean you should be mean to me.” OR
“Just because I was mean to you, that does not mean you should be mean to me.”

In the first, what subject is "it" referring to? It's like the "they" in "they say you should wait thirty minutes before swimming after a meal." Ambiguous and airy. In the third, "that" is similarly vague.

The second is preferable, and to me, reads better both with "doesn't" and "does not".