  We explore possible cosmological consequences of a running Newton's constant
$ G ( \Box ) $, as suggested by the non-trivial ultraviolet fixed point
scenario in the quantum field-theoretic treatment of Einstein gravity with a
cosmological constant term. In particular we focus here on what possible
effects the scale-dependent coupling might have on large scale cosmological
density perturbations. Starting from a set of manifestly covariant effective
field equations derived earlier, we systematically develop the linear theory of
density perturbations for a non-relativistic, pressure-less fluid. The result
is a modified equation for the matter density contrast, which can be solved and
thus provides an estimate for the growth index parameter $\gamma$ in the
presence of a running $G$. We complete our analysis by comparing the fully
relativistic treatment with the corresponding results for the non-relativistic
(Newtonian) case, the latter also with a weakly scale dependent $G$.
