"hawk-like," by 1703, from hawk (n.) + -ish. Sense of "militaristic" is from 1965, from hawk in the transferred sense.
Entries linking to hawkish
hawk n.
c. 1300, hauk, earlier havek (c. 1200), from Old English hafoc (West Saxon), heafuc (Mercian), heafoc, "hawk," from Proto-Germanic *habukaz (source also of Old Norse haukr, Old Saxon habuc, Middle Dutch havik, Old High German habuh, German Habicht "hawk"), from PIE root *kap- "to grasp" (source also of Russian kobec "a kind of falcon"). Transferred sense of "militarist" attested from 1956, probably based on its opposite, dove.
-ish
adjectival word-forming element, Old English -isc "of the nativity or country of," in later use "of the nature or character of," from Proto-Germanic suffix *-iska- (cognates: Old Saxon -isk, Old Frisian -sk, Old Norse -iskr, Swedish and Danish -sk, Dutch -sch, Old High German -isc, German -isch, Gothic -isks), cognate with Greek diminutive suffix -iskos. In its oldest forms with altered stem vowel (French, Welsh). The Germanic suffix was borrowed into Italian and Spanish (-esco) and French (-esque). Colloquially attached to hours to denote approximation, 1916.
The -ish in verbs (abolish, establish, finish, punish, etc.) is a mere terminal relic from the Old French present participle.