If your hu*****and is present, he might told the nurse that he is the father when you are too weak to argue with him. Then you will have to go to the trouble to change the baby's information later.
Your hu*****and might able to go to court, claim that he support you and your baby. However, he will not able to alaim to be the biological father unless you agree to do a DNA test. Certainly you will not agree the test if you do not want he be the father. The court could not force the DNA test against your will, because it is a civil dispute.