Coalition formation is a problem of great interest in AI, allowing groups of autonomous, rational agents to form stable teams. Furthermore, the study of coalitional stability concepts and their relation to equilibria that guide the strategic interactions of agents during bargaining has lately attracted much attention. However, research to date in both AI and economics has largely ignored the potential presence of uncertainty when studying either coalitional stability or coalitional bargaining. This paper is the first to relate a (cooperative) stability concept under uncertainty, the Bayesian core (BC), with (non-cooperative) equilibrium concepts of coalitional bargaining games. We prove that if the BC of a coalitional game (and of each subgame) is non-empty, then there exists an equilibrium of the corresponding bargaining game that produces a BC element; and conversely, if there exists a coalitional bargaining equilibrium (with certain properties), then it induces a BC configuration. We thus provide a non-cooperative justification of the BC stability concept. As a corollary, we establish a sufficient condition for the existence of the BC. Finally, for small games, we provide an algorithm to decide whether the BC is non-empty.