1876, "lacking an appetite;" see anorexia + -ic. The immediate source or model is perhaps French anorexique. As a noun meaning "person with anorexia nervosa" it is attested from 1913.
Entries linking to anorexic
anorexia n.
1590s, "morbid want of appetite," Modern Latin, from Greek anorexia, from an- "without" (see an- (1)) + orexis "appetite, desire," from oregein "to desire, long for," literally "reach out (one's hand), stretch oneself, stretch out for" (from PIE root *reg- "move in a straight line") + abstract noun ending -ia. In current use, often short for anorexia nervosa.
-ic
Middle English -ik, -ick, word-forming element making adjectives, "having to do with, having the nature of, being, made of, caused by, similar to," from French -ique and directly from Latin -icus or from cognate Greek -ikos "in the manner of; pertaining to." From PIE adjective suffix *-(i)ko, which also yielded Slavic -isku, adjectival suffix indicating origin, the source of the -sky (Russian -skii) in many surnames. In chemistry, indicating a higher valence than names in -ous (first in benzoic, 1791).
In Middle English and after often spelled -ick, -ike, -ique. Variant forms in -ick (critick, ethick) were common in early Modern English and survived in English dictionaries into early 19c. This spelling was supported by Johnson but opposed by Webster, who prevailed.