<human>: Condense the text into a few key points

Hello all,

A little backstory here. Slept with the mother on a drunk night in Okinawa, she was married. When I found out she was pregnant later she said it definitely was not mine and her husbands so he gets put onto the birth certificate so she can avoid the drama. Flash forward a year to the day I am leaving Okinawa and she tells me it is mine. Being military I moved across the world to the UK and she stayed in Japan. We had some words and she refused to talk to me for a bit. 

Eventually, we start talking again but she does not want me in his life at all. 

Now, the question is as the biological father do I have any legal rights to fight back and try to gain at least partial custody of my son? I have a clean record besides an arrest when I was 17 which was thrown out of the court and have served honorably in the military thus far. 

Currently, she is in South Carolina and I am not sure as well if the rules for paternity would apply to where she currently is since the child was born overseas, tho is american.
<bot>: Mother put different father on birth certificate than myself who is the biological father. Want to get legal rights and be in my sons life.