Why doesn't Batman use magic? I can understand he has a thing about guns, but it doesn't stop him employing other weapons. He has dedicated much of his life to learning seemingly endless skills to aid him in his battles against crime and evil, scientific, technical, physical, fighting, psychological, criminology, escape techniques, languages and so on. He creates and deploys endless gadgets, tech, and weapons - computers, Batmobiles, planes, motorcycles, bat armor, batarangs and other explosives, tasers, lasers, knockout gas, freezing gas, acid, robots, etc.
He's also well aware of magic, has often fought against it, and has some experience of using it occasionally. He hangs out with heroes who use it, so it's not like he thinks it is inherently evil. He's friends with some like Zatanna, and trained under Zatara (at least in some stories).
So why not regularly use magic? Just as he is not exactly Reed Richards or Tony Stark in the science and engineering worlds, I'm not suggesting he tries to become a Dr. Fate or Sorcerer Supreme level magic user. I mean learning some basic stuff or using some artifacts that could save his life or help take down a villain like his tech does. He has the intellect and dedication to learn crazy skills, and there are enough low level magic users around that make me think it wouldn't be that difficult.