"Corporations cannot attack conservatives and expect us not to fight back," Cagle said.
The news comes as Delta appeared close to convincing lawmakers to restore a lucrative sales tax exemption on jet fuel.
That proposal is part of Gov. Nathan Deal's larger tax overhaul, which has passed the House and awaits Senate input.
Delta said in a news release that the company's decision "merely confirmed its neutral status" in the national debate over guns.
The Associated Press contributed to this report.