I think that the explanation on this site is pretty clear myself.
Anthony did NOT originally come from the Greek word, and the original
Roman name it is derived from was not spelled with the "h". The "h" was added when people wrongly concluded it was derived from the Greek word for flower, but that was a mistake.
So no,
Anthony does NOT actually originate from the Greek word, though the modern spelling is derived from the mistaken idea that it did.