Stewart was stopped by a lone officer who responded to the call for help. Carthage Police Officer Justin Garner shot Stewart in the chest. Garner was also injured in the leg.
Stewart's relatives say he was targeting his estranged wife, who worked at the facility.
Prosecutors made the death penalty announcement at a hearing in Moore County on Thursday.
Stewart was present, along with the family members of some of his alleged victims.
Authorites say Stewart still has a bullet fired by Officer Garner in his body. He may have surgery to remove it.
Unlike his mugshot - where he appears with an unkempt beard and hair - Stewart was in court looking neater Thursday with his hair and bear trimmed.