Multiplicative subgroups are not restricted sumsets

We determine exactly which proper multiplicative subgroups of a prime field can be represented as a restricted sumset of the form $A\mathbin{\widehat{+}} A=\{a+a':a,a'\in A,\ a\ne a'\}$. We prove that a proper multiplicative subgroup $H\le\mathbb F_p^*$ cannot satisfy $H=A\mathbin{\widehat{+}} A$ whenever $|H|\ge7$, and that this threshold is sharp. In fact, such a decomposition exists precisely when $|H|\in\{1,3,6\}$, and we classify all decompositions in these exceptional cases. This gives a sharp, complete resolution of the restricted-sumset analogue of the generalized S\'ark\"ozy conjecture over prime fields. This significantly extends and refines previous results of Shkredov and Yip.