Legal Question in Family Law in Virginia

Yes, depending upon the policy of the hospital handling your maternity care, you may be able to leave the name of the father off the child's birth certificate, but that in no way will assure that you will have sole custody of the subject child should the matter of custody be contested.

And, yes, once paternity is legally established for this child, you will be

legally entitled to pursue child support from the father irrespective of whether or not you've been awarded sole custody.