From Greek 'kauteriazein' (to brand with a hot iron), from 'kaiein' (to burn) — all about burning at the root.
To burn the skin or flesh of a wound with a heated instrument or caustic substance, to stop bleeding or prevent infection.
From Old French 'cauteriser,' from Late Latin 'cautērizāre,' from Greek 'kautēriázein' (to burn with a branding iron), from 'kautērion' (branding iron), from 'kaíein' (to burn). The Greek verb 'kaíein' derives from PIE *keh₂w- (to burn), which also produced Latin 'cauma' (heat of the sun) and Lithuanian 'kūlė' (fever). The medical practice of cauterization is ancient, described in the Hippocratic corpus and practiced across the ancient Mediterranean
The word 'holocaust' is a distant relative of 'cauterize.' Greek 'holókauston' (a whole burnt offering) combines 'hólos' (whole) with 'kaustós' (burnt), from the same verb 'kaíein' (to burn) that produced 'kautērion' (branding iron) and hence 'cauterize.' Both words ultimately trace to the same PIE root for burning.