'non-idolatrous' is formed by the prefix 'non-' (meaning 'not') added to 'idolatrous'. 'Idolatrous' ultimately derives from 'idolatry' + the adjectival suffix '-ous'.

'idolatry' comes from Late Latin 'idolatrya', from Greek 'eidōlolatría' (where 'eidōlon' meant 'image' and 'latreía' meant 'worship'); 'idolatry' entered Middle English and produced the adjective 'idolatrous', and the modern formation 'non-idolatrous' is a more recent negative formation using the productive prefix 'non-'.

Originally referring specifically to the worship of images ('worship of idols'), the sense broadened to excessive devotion or reverence to people or things; 'non-idolatrous' therefore means 'not given to that worship or excessive reverence'.