No. The birth of a child is not a recognized qualifying event for a furlough or any other form of temporary release in the federal or state correctional system.
Furloughs are granted under an extremely narrow set of circumstances in the systems that offer them at all. A family member on their deathbed and attendance at a funeral of an immediate family member are the two situations most commonly considered. The birth of a child, as significant and emotional as it is, does not meet that threshold. The system's position is that childbirth is not an emergency requiring the inmate's presence in the way that an imminent death does.
It is a hard reality for families to accept, but it is consistent across virtually every correctional system in the country. Pregnant inmates themselves do not get released to give birth. They deliver in an outside hospital under guard and return to the facility. The standard does not bend for the birth of a child regardless of the circumstances.
What is sometimes possible, depending on the facility and the inmate's conduct record, is arranging a special phone call or video visit around the time of the birth. That is not guaranteed either, but it is far more achievable than a furlough. The request needs to go through the inmate's case manager or counselor well in advance of the due date, not at the last minute.
Once the baby arrives, sending photos through InmateAid gets them into his hands within about a week. Having a picture of his child is something he can keep and hold onto until he is home.