late 14c., "morally upright, righteous  in the eyes of God" ("Now chiefly as a Biblical archaism" - OED); also "equitable, fair, impartial in one's dealings;" also "fitting, proper, conforming to standards or rules;" also "justifiable, reasonable;" from Old French juste "just, righteous; sincere" (12c.) and directly from Latin iustus "upright, righteous, equitable; in accordance with law, lawful; true, proper; perfect, complete" (source also of Spanish and Portuguese justo, Italian giusto), from ius "a right," especially "legal right, law" (see jurist; from Latin ius also come English jury (n.), injury, etc.).

From c. 1400 as "right-minded, good in intention;" from early 15c. as "legal, lawful, right in law." Also "exact, precise; marked or characterized by precision; having correct dimensions" (late 14c.); of narrations, calculations, etc., "accurate, correct" (early 15c.). The sense in music, "harmonically pure, correct, and exact" is by 1850.

The more mundane Latin law-word lex covered specific laws as opposed to the body of laws. The noun meaning "righteous person or persons; Christ" is from late 14c. (The neuter adjective in Latin was used as a noun, iustum, "what is right or just").

mid-12c., "the exercise of authority in vindication of right by assigning reward or punishment;" also "quality of being fair and just; moral soundness and conformity to truth," from Old French justice "justice, legal rights, jurisdiction" (11c.), from Latin iustitia "righteousness, equity," from iustus "upright, just" (see just (adj.)).

Justice is the end of government. It is the end of civil society. It ever has been and ever will be pursued until it be obtained, or until liberty be lost in the pursuit. ["The Federalist," No. 51]

Meaning "right order, equity, the rewarding to everyone of that which is his due" in English is from late 14c. The Old French word had widespread senses including also "uprightness, equity, vindication of right, court of justice, judge." In English c. 1400-1700 sometimes also with a vindictive sense "infliction of punishment, legal vengeance." As a title for a judicial officer, c. 1200. Justice of the peace first attested early 14c. To do justice to (someone or something) "deal with as is right or fitting" is from 1670s. In the Mercian hymns, Latin iustitia is glossed by Old English rehtwisnisse.