“I was dating a woman for a short period when she became pregnant. If I do not sign the birth certificate or a paternity agreement, do not take a paternity test, and she does not pursue child support through the courts can the woman years later hold me liable for child support from the date of the birth?”
She would not be able to hold you liable from the date of birth unless she filed an action. In that case, the court could go back to the date of filing to render its order. Additionally, most states have provisions to go back years to collect for public assistance reimbursement. So if she gets any kind of medical aid or aid for dependent children, they can try to track you down and make you pay for past years.
My advice is (if you are sure this is your child) to pay her some amount you are willing to pay, so that she never does file an action or go on aid. Have a relationship with the child and help s/ he to grow up to be a happy healthy adult. Put the child’s best interests before your own.