In this paper, we develop a comprehensive and tractable analytical framework based on stochastic geometry to evaluate the performance of large-scale fog-aided device-to-device (F-D2D) networks with opportunistic content multicasting. As a part of the analysis, to resolve the contentions of file requests from the cache-incapable conventional user equipments (C-UEs), two simple yet typical candidate file selection schemes for cache-enabled fog user equipments (F-UEs), namely the random file selection (RFS) scheme and the most requested file selection (MRFS) scheme, are considered. Further, to suppress the harmful interference among the concurrent transmissions of F-UEs, a multicast-based opportunistic content delivery strategy is proposed by exploring the idea of opportunistic spectrum access (OSA). Assuming decentralized probabilistic caching, we first derive the activation probability of the F-UEs. Then, by adopting an appropriate approximation, the cache-hit probability, the coverage probability, and thereby the successful content delivery probability (SCDP) of the F-D2D network are evaluated. We also develop an iterative algorithm based on the gradient projection method to obtain a suboptimal caching policy for the maximization of SCDP. Extensive simulation and numerical results are presented to verify our analysis and demonstrate the superior performance of the proposed multicast-based opportunistic content delivery strategy.