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This paper introduces a novel stochastic multi-objective optimization framework for the integration of gravity energy storage (GES) with renewable resources—photovoltaic (PV) and wind turbine (WT)—in distribution networks incorporating demand response (DR), addressing key gaps in uncertainty handling and optimization efficiency. The GES plays a pivotal role in this framework by contributing to a techno-economic improvement in distribution networks through enhanced flexibility and a more effective utilization of intermittent renewable energy generation and economically viable storage capacity. The proposed multi-objective model aims to minimize energy losses, pollution costs, and investment and operational expenses. A new multi-objective enhanced weighted average algorithm integrated with an elite selection mechanism (MO-EWAA) is proposed to determine the optimal sizing and placement of PV, WT, and GES units. To address uncertainties in renewable generation and load demand, the two-point estimation method (2m + 1 PEM) is employed. Simulation results on a standard 33-bus test system demonstrate that the coordinated use of GES with renewables reduces energy losses and emission costs by 14.55% and 0.21%, respectively, compared to scenarios without storage, and incorporating the DR decreases the different costs. Moreover, incorporating the stochastic model increases the costs of energy losses, pollution, and investment and operation by 6.50%, 2.056%, and 3.94%, respectively, due to uncertainty. The MO-EWAA outperforms conventional MO-WAA and multi-objective particle swarm optimization (MO-PSO) in computational efficiency and solution quality, confirming its effectiveness for stochastic multi-objective optimization in distribution networks. Full article