Old English ceowan "to bite, gnaw, chew," from West Germanic *keuwwan (source also of Middle Low German keuwen, Dutch kauwen, Old High German kiuwan, German kauen), perhaps from PIE *gyeu- "to chew" (source also of Old Church Slavonic živo "to chew," Lithuanian žiaunos "jaws," Persian javidan "to chew").
Figurative sense of "to think over" is from late 14c.; to chew the rag "discusss some matter" is from 1885, apparently originally British army slang. Related: Chewed; chewing. To chew (someone) out (1948) probably is military slang from World War II. Chewing-gum is by 1843, American English, originally hardened secretions of the spruce tree.