if they came up to you, you minding your own buisness and called you out on your beliefs, then that makes them ignorant pricks, and if they can't see that they are doing what they consider 'wrong' then they deserv a sharp awakening to that fact, you did NOTHING WRONG, you didn't break the law or any 'law' of ethics or anything like that, you defended yourself and they dishonestly claimed (i assume i'm right here) that you slapped her, that puts her even more in the wrong, not to mention they only went there for an argument, and you had the right to finish you steak, because you paid for it (i assume it was just you and it wasn't on the house), and the only real reason that you were banned was because 9/10 in life, if a woman claims you hit her, their word will be classed as gospel (a saying for 'they won't be questioned on whether it's true or not, it will be assumed they are right'), and since you are 14, that makes it worse, the chef saw it as a woman telling him that some teen hit her, stereotypes make him not believe you, and there is no way that you can prove that you didn't do anything(unless the CCTV is over your table...).
here's what to do:
-go to the chef and ask him POLITELY if you can see the CCTV from the day of the incident, and if it records audio, then that's even better, but if not, get him to watch it through to prove that you didn't hit her,
-if it has audio, get him to listen to it as well to prove that they started on you,
- if both those go to plan, ask him to sleep on whether you would be allowed back in