This article proposes a data-driven modeling framework with physically consistent Gaussian Processes (GPs), enabling learning-based disturbance estimation for uncertain mechanical systems with covariance-adaptive Momentum Observers (MOs). We present novel error bound results in closed form, holding with high, exactly computable probabilities, and exploit the multidimensional, physically constrained function distribution induced by the differential equation structure of Lagrangian systems. The inherent uncertainty quantification provided by GPs and the derived model error bounds are then leveraged to probabilistically guarantee exponential stability of a class of data-driven, adaptive MOs with user-definable convergence parameters. We demonstrate the performance of our proposed methods in simulations and physical experiments, showing significant improvements compared to the state-of-the-art from industry and research.