The incident happened Thursday.
"After initial attempts by institution staff to stop the assault proved unsuccessful, the tower officer fired a warning shot which failed to end the assault," Heffington said. "Subsequently, the tower officer fired again and struck (Back)."
Medics took Back to a hospital in Huntington, W.Va., where he died Saturday.
Back was arrested and convicted of robbing banks in Kettering, Centerville, Dayton, Riverside and Hamilton in 2004 and 2005. The FBI listed him as one of their top 10 most wanted, and a profile on "America's Most Wanted " TV show eventually led to his arrest.
He was found guilty of the bank robberies in federal court and ordered to pay $128,109 in restitution.
Heffington said the other inmate involved in the stabbing was recently released from the hospital.
The FBI is investigating the incident.