No, the damage is being done by a yellow-bellied sapsucker, related to the woodpecker. This federally protected bird searches shrubs and trees with thin bark and pecks rows of holes in a distinct pattern. Sapsuckers feed on the sap that flows from the hole. Healthy wax myrtles can sustain a large number of holes without any apparent damage. You can apply an approved bird repellent to the trunks to discourage its activity.
Resources: (online)
Woodpeckers
