No, he can not do that. He is going to be required by the Court to pay child support, and if he fails to do so he will eventually be held in contempt of Court and placed in jail in order to induce compliance with the Court's order. If he has a job, ask the Court to issue a wage garnishment so that he doesn't have choice about whether he is going to pay child support. Best of Luck!
Technically he could surrender his rights without your permission, although most courts find this against public policy, unless there is someone in line to adopt such as a step-parent. Even if a court accepted a surrender or his rights were terminated he would be still responsible for the child support as only an adoption would sever his duty to support.
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