I believe it is entirely possible that the apparition is legitimate. If I recall correctly, it was after the apparition that large numbers of natives abandoned paganism and human sacrifice for Christianity. While I don't think that RCism is the true Church, I really doubt that it was likely that random natives in rural Mexico were going to find out about Orthodoxy in the 1530s so I'm willing to believe that the Virgin Mary took an "executive" decision to appear to them to convince them to stop human sacrifice and paganism even though they didn't end up Orthodox (step in the right direction, etc.).
Just my personal opinion, and not official Orthodox teaching. I have seen the icon of Our Lady of Guadalupe in one OCA bishop's home though, for what it's worth.
I am inclined to agree. Especially when one considers how horrific the Aztec religion was with the human sacrifices, more human sacrifice in the ritualistic Flower Wars, in which the Aztecs always won and the weakened captive civilizations of Southern Mexico always lost, and other evils. Now, imagine if the Aztecs had retained their religion, and eventually overpowered the Spanish, which happened anyway but under Catholicism. But the Aztecs were technologically impressive; Cortes and his party record Tenochitlan as being as impressive a city as they had seen. So despite the accidental devastation of small pox, they could have recovered, and by the late 18th or early 29th century thrown off the Spanish yoke at the time the Spanish Empire was crumbling, perhaps aided by shortsighted Dutch merchants. Then they might well have tried to revive the human sacrifices to thank the Gods, and perhaps they would have sought victims from elsewhere in the Americas, especially the former Spanish Empire, or even Europe.
This unpleasant alternate reality did not happen because of this one image, which was enough to smash Aztec paganism. Only a substrate of it remained in unpleasant cultural events like The Day of the Dead, which were unfortunately eventually tolerated, and were largely the fault of dubious practices in the Roman faith (even outside of Mexico, All Souls Day celebrated according to the Tridentine rubrics can be rather morbid, with the empty coffin, and the pall, sometimes with an actual skull and crossbones decorating it; look at Fr. Z's blog at wdptrs.com/blog for an example he used last year and was rather proud of). This substrate has unfortunately become a fertile breeding ground for fungal growths from spores ejected by the dyimg Aztec faith. I would cite the cult of Samta Muerte, and the related religious rites, that have Amomg other things come to play a huge role in the so called Narco Culture of the drug gangs which have killed so many Mexicans just since 2008 and devastated tourism in Tijuana and other once popular destinations.
But the majority of !exicans are opposed to this, and have in my experience high moral standards. And the reason for that is their devout Catholicism. And the reason for that is Our Lady of Guadalupe. Which by the way, is an entirely realistic depiction; it is not outside the realm of possibility to say the Theotokos looked like that.
Now, stylistically, it's not my favorite icon, but in terms of the beneficial effects it had, literally creating the Mexican nation of devout Catholics; people of indigenous and Spanish descent who eventually unified on the basis of faith, and eradicating the unsavory aspects of the Aztec nation, that being its religion and warlike nature, thus giving birth to Mexico, as we know it, it could well be my favorite culturally. I love Mexico, pious Mexicans, the hardships they've endured and their deep piety. There is a store in Tijuana, not one of the bartering places selling junk at negotiated prices, but rather a fixed price speciality shop, called a Hand Arts in English, where one may find the finest, most exquisite Mexican embroidery and lacework, in delicacy exceeding that of Europe. Or there was assuming the drug war did not put them out of business. And the Hotel Cesar, where I once dined, which features exquisite cuisine and where the first, and in my opinion best, Caesar salad is made, table side, using an ingredient illegal in the US: raw eggs. But I had no indigestion. And cities like Durango and Chihuahua have picturesque "downtowns" (the word hardly seems adequate) that look precisely like the most charming cities in Spain. And all this beauty was made possible by this icon, which bridged the gap between two warring cultures and allowed for a third, new culture to arise like a mythical Phoenix from their ashes. What Mexico needs now is another icon of the Virgin Mary of similiar miraculous origin, or another providential event, to end the drug wars and wipe out the residual, toxin-spewing remnants of Aztec religion which have become so intimately connected to the Narco Culture, something to call the gangs of Mexico and the corrupt politicians and police to repentance, so that the decent ordinary folk may live a normal life of piety.