Surgeon admitting death leads to terribly acrimonious feud (8)
I believe the answer is:
vendetta
'feud' is the definition.
(vendetta is a kind of feud)
'surgeon admitting death leads to terribly acrimonious' is the wordplay.
'surgeon' becomes 'vet' (veterinary surgeon).
'admitting' means one lot of letters goes inside another (admit can mean to welcome in or accept).
'death' becomes 'end' (synonyms).
'leads to' says to take the initial letters.
The first letters of 'terribly acrimonious' is 'ta'.
'vet' enclosing 'end' is 'vendet'.
'vendet'+'ta'='VENDETTA'
(Other definitions for vendetta that I've seen before include "Violent feud" , "Unremitting battle" , "Bitter feud between families" , "A prolonged feud, associated with Sicily" , "Family blood feud" .)