Where did the phrase "to eat crow's pie" originate?

There might be some mixing of sayings here to express humility, which could be separated into 'eating crow' and 'eating humble pie'. A crow is pretty bony and has virtually very little in the way of meat, so yeah, eating it would be akin to eating humble pie.
http://idioms.thefreedictionary.com/eating+crow

Actually, umble pie was regarded by many as a delicacy (I suppose in the same way some people regard tripe as one). Note 'umble', not 'humble'. There is a real dish called umble pie. The connection with 'humble' is folk etymology at work. Rooks were cooked in pies, and also crows, but these were cheap, labourers' food (until certain very expensive restauranteurs decided to revive them...). In Lewis Carroll's (rather neglected) 'Sylvie and Bruno', Bruno says sadly that his dinner had been 'a little bit of a crow', but his older sister Sylvie translates this into rook pie for the benefit of Mister Sir. The expressions 'eat crow' and 'eat humble pie' are separate, and eating crow pie is a hybrid. By the way, there is no basis for the story of the white hunter and the Native American hunter, or any of the other variations of it. That story is just one of those legends that we like to debunk here.