No. The father can never avoid his obligation to pay child support by "giving up his rights". The only way to avoid a support obligation is if the child is adopted by another man, in which case you would no longer be considered the father of the child, and no longer have an obligation of support.
Mr. Rosen practices matrimonial and family law in Great Neck, NY. His response here is not legal advice and does not create an attorney/client relationship. The response is in the form of legal education and is intended to provide general information about the matter in question. Many times the questioner may leave out details which would make the reply unsuitable. You are strongly advised to confer with an attorney in your own state to acquire more information suited to your particular circumstances.