You are encountering the clichés of romance fiction, where the man is almost invariably depicted in a superior position, not only because he is likely to be taller than his female companion. The adverb down accentuates first this superiority, the proximity to each other, and finally their intimacy, since only when one enters the personal or intimate space of another does a difference in height become so apparent it warrants an adverb. The down and up of it, however, is generally only of interest to this genre of fiction.
Staring, gazing, or looking into someone's eyes, however, is something real people do in real situations, especially intimate ones. One can gaze at the sky for any number of reasons, but gazing into the sky suggests a moment of reverie — a word also favored by this genre — as if to penetrate toward some deeper meaning. Looking at someone's eyes suggests, say, that they are a striking color or shape but would convey only interest, not intimacy or physical proximity.