Cheater of a father threatens to claim full custody of my baby

I found out my now ex-fiancé cheated on me after finding out I'm expecting. He's stationed overseas, and I told him that I never want him to see the baby (once it's born) because he has A) physically abused me before, and B) proven himself unworthy of being a good father through his actions. He told me he would spend every penny he has to take me to court to get full custody of the baby and because he has income & I don't (I'm a dependent), he'll win the case. Is this true?! I read into it, but the most I know is that if he's not present at the time of baby's birth/not there to sign baby's birth certificate, I will assume all parental rights. Has anyone had to deal with this or has any advice on how to deal with it?