Insult grasping private refuses to acknowledge (7)
I believe the answer is:
disowns
'refuses to acknowledge' is the definition.
(to disown is to refuse to acknowledge as one's own)
'insult grasping private' is the wordplay.
'insult' becomes 'diss' (diss can mean to insult or disrespect).
'grasping' indicates putting letters inside.
'private' becomes 'own' (as in one's own or private belongings).
'diss' going around 'own' is 'DISOWNS'.
(Other definitions for disowns that I've seen before include "Has nothing to do" , "Refuses to acknowledge" .)