Caramanlis said Defense Minister Spilios Spiliotopoulos had asked for Lietenant General Panayotis Papanikolaou's resignation.
He said he had declined Spiliotopoulos' offer to also step down over the incident.
"There was no political responsibility but a breakdown in the army system," he said.
The Chinook helicopter carrying the Greek Orthodox Patriarch of Alexandria, Peter VII, crashed Saturday on its way from an Athens military base to the Mount Athos monastery near the Aegean Sea.
Opposition newspapers on Sunday said rescue workers were slow to respond to the accident.