Do Immigrants have the same rights with child custody/visitation as a citizen?

Both myself and my child are US citizens. I married an H1B (and made it clear he wouldn't get a GC thru me) altho he thought I would change my mind I did not. Now he is trying thru his work for his GC. My question is, if I divorce him do I have to share my child with him. My child does not know him, and he has never lived with us. My concern with visits is he has threatened to abduct my baby to his home country of India (if I divorce him, he umes he'd get visitation & could then run) I have talked with 2 lawyers (but finances keep me from shopping around for different lawyer opinions) both lawyers have stated he would get unsupervised visits (which terrify me) unless/until he does something (like steal my child) my point & fear is by that point it's WAY to late...and I cannot take such a risk. I have stayed married to him for only that reason. As long as we are married I am in control, if we divorce the courts will take control. Am desperately trying to protect my 3 yr old.