Green’s Dictionary of Slang

(US) Caucasian.

1966		W. Wilson LBJ Brigade (1967) 37: Nobody is happy except the command officers, they hoard the round-eyed prostitutes.
1970		Current Sl. V:1 18: Round eye, adj. Caucasian.
1977		‘John le Carré’ Honourable Schoolboy 136: We’d tell them how you play roundeye musical beds at the young bankers’ club.
1977		L. Heinemann Close Quarters (1987) 285: Round-eyed pussy walking the streets just like back in Des Moines.
1981		M. Baker Nam (1982) 163: The GIs called nurses roundeye tail.
1985		(con. 1969–70) D. Bodey F.N.G. (1988) 170: Hey, fuckers. Nuthin’ but round-eyed pussy here.
1990		(con. c.1970) G. Hasford Phantom Blooper 31: I constantly reassure those who fear you because you have blown away a round-eyed Marine.