Not necessarily. Egg moves are an entirely different concept that isn't directly determined by egg groups, egg groups are just to find which Pokemon are compatible. The link below shows the egg moves that Charmander can get; the ones it can inherit from Gyarados are Bite, Crunch and Dragon Dance. So if you want Charmander to inherit a Dragon move from Gyarados, your only option is Dragon Dance. http://pokemondb.net/pokedex/charmander/egg
BTW I moved your answer to a comment as it was more appropriate there, I hope you do not mind.