  We have recently presented a manifestly local and general coordinate
invariant formulation of a nonlocal approach to the cosmological constant
problem. In this article, we investigate quantum effects from both matter and
gravitational fields in this formulation. In particular, we pay our attention
to the gravitational loop effects and show that the effective cosmological
constant is radiatively stable even in the presence of the gravitational loop
effects in addition to matter loop effects. For this purpose we need to add the
$R^2$ term and the corresponding topological action as an total action, which
should be contrasted with the work by Kaloper and Padilla where the topological
Gauss-Bonnet term is added instead of the $R^2$ term. The advantages of our new
formulation compared to that by Kaloper and Padilla are that not only we do not
have to assume the scale invariance which is required to render the space-time
average of the square of the Weyl curvature vanishing, but also our formulation
would lead to the $R^2$ inflation in a natural manner.
