Should I ignore him instead of arguing back?
Could you explain why you want to ignore him?
Because I cannot win the argument
It is wrong and rude to ignore people. Just because you cannot win the argument does not mean that you should not respond. Be polite in all situations.
I'm not going to stop ignoring him, if you try to fight back you lose and I'm a bitch for winning.
The best option is neither ignoring them nor arguing with them. Rather, make an effort to have a civil discussion and discuss the conflict maturely and in a way that is most likely to remedy the problem. It's not always about winning and losing, and furthermore, it's okay to be wrong sometimes.