Green’s Dictionary of Slang

gomer n.3

[? abbr. get out of my emergency room; note Doug Wilson posting to American Dialect Society List 21/7/01: ‘The acronym-etymology is spurious, I think; another bogus candidate is “Grand Old Man of the Emergency Room”. I think the origin is related to the other “gomer” – like in “Gomer Pyle” [...] maybe the original form was “old gome” or so, and maybe “gome” = “man” (from OE “guma” or so; cf. “gome” [and “gomerel”] in OED) [...] Probably reinforced [...] by “gummer” = “one who gums (i.e., chews without having any teeth)” and maybe by “gnome”.’]

an old, dirty, difficult or chronically ill hospital patient, usu. male; thus gomere, the female equivalent.

[US]National Lampoon July 76: Gomer. A senile, messy or highly unpleasant person.
[US]S. Shem House of God (1983) 38: If I’m not mistaken, it’s from one Ina Goober, whom I admitted six times last year. A gomer, or rather, the feminine, gomere.
[US]N.Y. Times Mag. 9 Nov. 16: A gomer [...] is a patient [...] who is whining and otherwise undesirable.
[US]S. Hicks ADS-L 🌐 27 July My wife works in a small-city midwestern hospital. The term ‘gomer’ in her work-culture is applied to ‘trailer trash’ .... lower-class white yokels with no jobs and/or medical insurance, who use the local emergency-room as their primary health-care service (even for minor wounds, most inflicted by other family-members, or through alcohol-abuse, or both). ‘Gomer’ in that usage is popularly said to be an acronym for ‘Get Out of My Emergency Room.’.