Aylmer is clearly the main character of this tale and the one around whom the tale revolves. His desires drive the story; his mistakes embody Hawthorne's didactic end-game. As a reader, you may sympathize with Aylmer's intentions, which he truly believes to be good. In his mind, and in Georgiana's, his love is all the more noble, and all the more lofty, for his dedication to perfection.
If you like Aylmer as the antagonist, you may be tempted to assign Georgiana is the protagonist. She is wronged by the antagonist (her husband), she is basically innocent, and she suffers on no account of her own, which means our sympathies lie with her. If you tend to read "The Birthmark" as a tragedy, then you may want to go with this selection for protagonist.