Answer: No, martial arts does not stunt growth.
For some reason this myth never goes away. There is no evidence to suggest that any form of exercise stunts a child's growth.
Growth depends almost exclusively on genetics and diet.
I suppose that you could make the argument that serious injury does cause stunted growth, particularly if the injury is to the lower back. Such injuries may be caused by landing incorrectly, having a lock put on incorrectly or too hard, aggravating a previous injury, etc. Not every injury can be prevented, but the likelihood of them happening can be minimised under the supervision of a competent instructor.Where does this claim come from?
Tracking down the origin of such a claim is difficult. Similar untrue claims exist in weight training and other more intensive sports. Usually this type of claim originates by someone picking up a piece of research and misunderstanding its context. More specifically, I would guess that there was a case in which a child's growth was stunted because of an injury, but the claim was generalised to suggest that all forms of intense exercise stunt growth.