Doctor said baby had a greenstick fracture. Do they normally put a cast on this type of break?

Age/mobility/need. Splinting often depends on where the fracture is. If it is the collar bone, the most often fractured bone in the body, sometimes a newborn baby is just left to heal. If it is an extremity and the baby is old enough to move around & get into trouble, it might be splinted for comfort or protection.