This paper proposes a novel robust decentralized charging strategy for large-scale EV fleets. The system incorporates multiple EVs as well as inelastic loads connected to the power grid under power flow limits. We aim at minimizing both the overall charging energy payment and the aggregated battery degradation cost of EVs while preserving the robustness of the solution against uncertainties in the price of the electricity purchased from the power grid and the demand of inelastic loads. The proposed approach relies on the so-called uncertainty set-based robust optimization. The resulting charge scheduling problem is formulated as a tractable quadratic programming problem where all the EVs' decisions are coupled via the grid resource-sharing constraints and the robust counterpart supporting constraints. We adopt an extended Jacobi-Proximal Alternating Direction Method of Multipliers algorithm to solve effectively the formulated scheduling problem in a decentralized fashion, thus allowing the method applicability to large scale fleets. Simulations of a realistic case study show that the proposed approach not only reduces the costs of the EV fleet, but also maintains the robustness of the solution against perturbations in different uncertain parameters, which is beneficial for both EVs' users and the power grid.