I'm 7 months pregnant and since I was 2 months my boyfriend began to pick fights with me and leave the house. He became verbally abusive towards me so when I was 3 months he finally left and has not come back. However, the baby had a problem with his growth I called him with my concern and to tell him everything that was going on. He did not answer the phone (I left a voicemail). I sent him an e-mail he did not respond. I sent him a text message and he responded with curse words asking me what I wanted. I explained that there was a problem with the baby and he did not respond. Since that day I have not spoken to him. I plan to not put his name on the birth certificate nor give the baby his last name. Can he still have rights to my son? If I have someone that will be willing to adopt him as his own can he still have rights to my son? I don't believe he loves him and he might want to use him to get back at me or as a pawn to upset me. I'm also concerned because he is here in the US on a work permit, he has no bank account owns no property and can leave to his country with no problem and my fear is that he will try to take my son form me. What can I do?

For him to have any rights without being on the birth certificate he would have to do a paternity test. With his name not on the birth certificate you would not be able to go after him for child support.

I do not want nor do I need child support

... I plan to not put his name on the birth certificate nor give the baby his last name. Can he still have rights to my son? ... What can I do?
He can establish paternity by filing a petition in court. Adoption law varies from state to state; it is within the context of an adoption petition that the father's rights are terminated.