Likely from 'course' in the sense of 'ordinary' — coarse cloth was everyday cloth, with spelling diverging in the 1600s.
Rough or loose in texture; lacking refinement or delicacy; crude or vulgar.
Of uncertain etymology, appearing in Middle English as cors, corse meaning ordinary, common, rough in texture. The most accepted derivation is from the phrase in course (in the ordinary run of things), from Old French cours, from Latin cursus (a running, a course), from currere (to run), from PIE *kers- (to run). Under this theory, something coarse was
'Coarse' was originally just a spelling variant of 'course' — 'coarse' bread was bread 'of the ordinary course,' meaning everyday bread as opposed to the fine white bread eaten by the wealthy. The two words were not distinguished in spelling until the 17th century. This is why 'coarse' and 'course' sound identical — they were once