I think that one way to approach this would be to look at the similarities between this excellent play and other Greek tragedies, such as Antigone or Oedipus Rex. The Irish influence is obvious in the setting and the way in which Catholic superstition is so evident. However, the way in which Maurya in particular has to battle to accept her fate and destiny, and the dignity with which she does this, is comparable to Greek tragic heroes at their best. Note one of her final speeches and the quiet dignity that Synge bestows upon her character:
They're all gone now, and there isn't anything more the sea can do to me... I'll have no call now to be up crying and praying when the wind breaks from the south, and you can hear the surf is in the east, and the surf is in the west, making a great stir with the two noises, and they hitting one on the other.
Maurya is truly a tragic hero in the way that she faces the vicissitudes of life and still manages to greet them with dignity and self-respect.