1570s, "aromatic resin used for healing wounds and soothing pains," from Latin balsamum "gum of the balsam tree," ultimately from Semitic (see balm). There is an isolated Old English use from c. 1000, and Middle English used balsamum. Originally in reference to Balm of Gilead, later extended to various other aromatic preparations from trees and shrubs. As a type of flowering plant of the Impatiens family, it is attested from 1741.
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.
<a href="https://www.etymonline.com/word/balsamic">Etymology of balsamic by etymonline</a>
Harper, D. (n.d.). Etymology of balsamic. Online Etymology Dictionary. Retrieved $(datetime), from https://www.etymonline.com/word/balsamic