"stem of a plant," early 14c., probably a diminutive (with -k suffix) of stale "one of the uprights of a ladder, handle, stalk," from Old English stalu "wooden part" (as of a harp), from Proto-Germanic *stalo; related to Old English steala "stalk, support," and steall "place" (see stall (n.1)).
"pursue stealthily," Old English -stealcian, as in bestealcian "to steal along," from Proto-Germanic *stalkojanan, probably from a frequentative of the root of steal (cf. hark from hear, talk from tell). Or it may be from a sense of stalk (v.1), influenced by stalk (n.). Meaning "harass obsessively" first recorded 1991. Related: Stalked; stalking.
A stalking-horse was literally a horse trained to allow a fowler to conceal himself behind it to get within range of the game; figurative sense of "person who participates in a proceeding to disguise its real purpose" is recorded from 1610s.
"walk haughtily" (opposite meaning of stalk (v.1)) is 1520s, perhaps from stalk (n.) with a notion of "long, awkward strides," or from Old English stealcung "a stalking," related to stealc "steep, lofty."