My first character was so buggy (one of his bridge officer disappeared, unable to train and had a hole in his chin area) that I decided to delete him. Lo and behold I try over and over and he cannot be deleted. I tried with my second character and she cannot be deleted either. (Nothing wrong with her, but I decided to check.
I think there must be an issue with the server. In closed beta I was able to delete characters perfectly fine. According to the login right now there is an issue with creating characters. It might be the same issue keeping you from deleting your character and me from retrieving mine.