We have recently presented a manifestly local and general coordinate invariant formulation of a nonlocal approach to the cosmological constant problem which has been proposed by Carroll and Remmen. In this article, based on our formulation, we investigate some quantum aspects of this approach to the cosmological constant problem. In particular, we explicitly evaluate quantum effects to the effective cosmological constant from matter fields at one-loop level. We show that the effective cosmological constant receives radiative corrections and is not stable against quantum corrections so that it must be fine-tuned at every stage in the loop expansion. Next, we propose a new geometrical model of a nonlocal approach to the cosmological constant problem, and we show explicitly that in this new model the effective cosmological constant is indeed stable against radiative corrections.