Things you can do with your eye(s)...
open, close / shut, lift / raise, cast, turn, screw up (against the glare of the sun), strain, blink, cross, roll, rub, lay/set on sth, protect, shade, shield, cast... but not throw, although I'd like to ask a native speaker to explain to me the difference between cast and throw...
(courtesy to Oxford dictionary of collocations)
If ever, under the pressure of all those unfortunate non-native users of English, the phrase "to throw an eye" should ever near "acceptable" or worse, "widespread", it will have earned its place on every editor's little shelf of horrors along with "she tossed her head"...
Don't throw about with your body parts please!
Instead, check a thesaurus -- entry for "look"
1 Mrs. Wright looked at him | she looked out of the window
GLANCE, gaze, stare, gape, peer, fix one's gaze, focus; peep, peek, take a look;
WATCH, examine, study, inspect, scan, scrutinize, survey, check, contemplate, consider; see, observe, view, regard, pay attention to, take note of, mark; glimpse, spot, spy, lay one's eyes on, catch sight of, eye, take in, ogle; informal take a gander, give someone/something a/the once-over, have a squint, get a load of, rubberneck, recce; Brit. informal take a dekko, take a butcher's, take a shufti, clock, gawp; N. Amer. informal eyeball; archaic behold, espy, descry.
1 Rachel glanced at him nervously
TAKE A QUICK LOOK, look quickly, look briefly, peek, peep; glimpse, catch a glimpse of; Scottish keek; informal sneak a look, take a gander; Brit. informal take a dekko, have a shufti, take a butcher's; Austral./ NZ informal squiz; archaic glance one's eye.
-opposite(s): GAZE, SCRUTINIZE.
(courtesy to Oxford Thesaurus)