Fatty liver can have quite a few causes. For most, abstinence and healthy food with as little fructose as possible will completely reverse it. But there can be other causes like insulin resistance, which is why talking to your doctor is your only safe bet.
However, continuing to drink with a fatty liver diagnosis is like playing Russian roulette.
FWIW I personally saw a complete reversal after 6 months sober combined with a low glycemic load diet. Alcohol and sugar are very similar when it comes to your liver, hence why even kids can have non-alcoholic fatty livers. But to be sure, consult your doc; it's their job and they have been trained in this matter.