As long as the father is the biological father then he will have full parental rights and could pick up the child. Even if the mother left the father off the birth certificate, the father could sue the mother for custody/visitation and the father’s name would likely eventually be added to the birth certificate.
The father cannot “sign over” his rights. The only way a parent’s parental rights can be terminated is by a court proceeding and a court order.
With the father being in the military, he may have to provide documentation to the military that his child has a primary custodian during the time of deployment in the event he is deployed. Otherwise, a custody case would proceed as normal and custody and visitation would be subject to his work schedule/military duties. That is of course assuming the father were to file a lawsuit against the mother for custody/visitation.
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