All the woman has to do is claim that the man she named is the father, and then deny any chance to do a DNA test. I personally know of two men that were stuck in just such a way.
https://dnatesting.com/legal-refuse-paternity-test/
What if the Mother or Father Refuses to Participate in the Paternity Test?
If the mother refuses
Chances are good the court will automatically order that, because paternity has not been definitively established, she is not entitled to receive child support from the alleged father.
Keep in mind that in most states, a possible father can only contest paternity if he does so
within a specific time frame, so,
if a woman refuses a paternity test, it’s extremely important for any man who questions paternity to request a test before the baby’s born or as soon as possible afterwards.
Because the court considers the welfare of the child over everything else, if a man is on the birth certificate and waits too long to test,
he may never be off the financial hook—even if a paternity test eventually proves he’s not the biological father.
If the father refuses
He may face legal consequences for his refusal, such as
being held in contempt of court: criminal charges could be filed or he could be fined. In many cases, the court may simply enter a judgment against the man and order him to pay child support.
