We present a stochastic geometric framework for gravity, starting from the Gravitational Balance Equations (GBE)~cite{lavenda} which arise from varying the Einstein-Hilbert action with respect to sectoral scale factors in a doubly-warped spacetime. The extrinsic curvature is promoted to a random field, and a moment hierarchy is derived from the GBE. A geometric projector closure maps second moments to an effective fluctuation curvature, yielding closed mean equations without ad-hoc stress tensors. The fluctuation energy obeys a generalized Bochner formula, linking geometric dissipation to the mean extrinsic curvature and the intrinsic curvature of the leaves. This approach provides a self-consistent probabilistic description of gravitational fluctuations, revealing that classical general relativity is not a fundamental deterministic theory but rather the first-moment truncation of a more complete stochastic geometric description. In particular, the so-called ``exact'' vacuum solutions of Einstein's equations--such as Schwarzschild--are not exact; they are mean-field approximations that neglect the essential nonlinear term (K_{AB}K^{AB}) and all higher fluctuations. This neglect becomes manifest in regimes beyond the photon sphere ((G<3M)), where the classical hierarchy of terms breaks down and the mean-field description yields unphysical results.