Under the same spirit of the Galileon-Horndeski theories and their more modern extensions, the generalized $SU(2)$ Proca theory was built by demanding that its action may be free of the Ostrogradski's instability. Nevertheless, the theory must also be free of other instability problems in order to ensure its viability. As a first approach to address this issue, we concentrate on a quite general variant of the theory and investigate the general conditions for the absence of ghost and gradient instabilities in the tensor sector without the need for resolving the dynamical background. The phenomenological interest of this approach as well as of the variant investigated lies on the possibility of building cosmological models driven solely by non-Abelian vector fields that may account for a successful description of both the early inflation and the late-time accelerated expansion of the universe.