At the start of the fil, Rocky is recovering in hospital from his fight with Apollo. A young boy asks for his autograph and Rocky duly signs it using his right hand - even though we know full well that he is a southpaw i.e. left-handed.
You DO have to be left-handed to be a Southpaw. That's what it means.
Also, if Rocky could write right handed, then he could box right handed, even if his left were stronger. Then he wouldn't have had the problems finding a fight, because "Nobody wants to fight a southpaw."
I don't think his left hand was broken, because he was clearly using that hand to hold the pad.
Also, a little boy didn't ask for his autograph. It was the nurse.