Absolutely. Under Georgia law, both parents are required to support their children financially. While a child support order establishes the amount that must be paid, the obligation to pay exists despite the non-existence of an order. If a father is not financially supporting his child, then it is possible for him to be charged with abandonment. [Keep in mind... filing does not automatically mean there will be a finding of guilty.]
An affirmative defense to the charge of abandonment is for the man to refute paternity. If you know, or even suspect, that the child is not yours, you have the right and opportunity to tell the court that you doubt paternity. The court can (and likely will) require a dna test before they will rule that you have abandoned the child. If it is determined that you are not the father, you should be free from having to deal with this issue again. If it is determined that you are the father, a support order will be established and issued.
~ Kem Eyo
The above answer is a general explanation of legal rights and procedures. It does not constitute legal advice. Nor does it establish an attorney-client relationship between the individual posting the question and the attorney providing the answer.
Speak to an attorney about requesting that the Court Order a paternity test and - if the test shows that you are the father - filing a petition to legitimate your child and seeking custody rights / parenting time. Make sure to talk to the attorney about your historical involvement with the child - how often you have communicated with and visited with the child. Your question does not indicate whether your child is an infant, toddler, young child or teenager. If a newborn (and even if an older child), let the attorney know if you attended prenatal visits, the child's birth, parenting classes, and if you have support the child financially in the absence of a court Order. Make sure to ask the attorney about what you need to demonstrate to the Court in order to have the legitimation granted (that you have not abandoned your opportunity interest and that the legitimation is in the best interest of the child). Ask the attorney to explain the different types of custody - physical custody, joint physical custody, legal custody, joint legal custody - and how custody is determined (ask about the Child Support Guidelines). Also, make sure that you discuss your obligation to support the child now - in the absence of a Court Order. My colleague is correct that under Georgia law, both parents have a duty to support their child. Best of luck to you.
This answer is offered for informational purposes only. It is not offered as nor does it constitute legal advice. This answer does not constitute an attorney-client relationship. Do not rely on this answer in prosecuting or defending against any criminal or civil legal action. Speak to an attorney in your area about how to protect yourself and your interests.