It's possible to dive without any training and live to tell the tale. But just because it's possible doesn't mean you should. The physiological stress being placed on the body is tremendous. Lack of knowledge, skills, and technique can lead to physical injury, including death. Without proper training, it's not a matter of if an injury will happen, it's a matter of when.
On the other hand, proper training can reduce the chances of injury by an amazing amount. In fact, according to data from the National Safety Council and the Divers Alert Network, choking on food poses a greater risk than certified divers diving.