My Ex-Husband was arrested for lewdness with a minor under the age of 14. The child was under our guardianship at the time of the offence. I was pregnant with his child when this took place and when he was convicted. He has informed me that he will fight for custody or visitation of our child. I currently have sole physical and legal custody of her. Can he be awarded either and is it possible for me to have his rights removed?

Considering the circumstances the most he is likely to get is supervised visitation and I think that is a long shot. You don't need to do anything until he actually goes to court. He is probably just blowing smoke.

Wait until he makes a move then consult an attorney.