Supply chain networks exhibit highly coupled dynamic behavior resulting from uncertain demand patterns, delayed information exchange, and decentralized operational decisions. These characteristics often lead to instability phenomena, most prominently the bullwhip effect, where small variations in customer demand become significantly amplified as orders propagate upstream through the network. This study introduces a geometric nonlinear control framework aimed at stabilizing multi-echelon supply chain systems operating under stochastic demand disturbances and structural delays. The inventory dynamics are reformulated within a nonlinear control-affine structure, allowing the use of structural controllability analysis together with Lyapunov-based stability verification. A feedback control law that combines integral regulation with a nonlinear damping component is proposed in order to attenuate oscillatory inventory behavior and limit variance propagation across supply chain stages. To assess the effectiveness of the proposed approach, a synthetic dataset is generated using an autoregressive demand model enriched with seasonal patterns and disruption events. A comprehensive Monte Carlo simulation framework is employed to evaluate system performance across multiple uncertainty scenarios, including demand shocks, transportation delays, and volatility changes. The numerical results reveal substantial improvements in dynamic performance. In particular, the proposed controller reduces the bullwhip amplification ratio by approximately 68%, shortens settling time by more than 60%, and significantly enhances robustness against delay-induced instability when compared with conventional linear feedback strategies. The findings demonstrate that nonlinear geometric control concepts can provide a rigorous analytical foundation for stabilizing complex supply chain networks and improving operational resilience in automated logistics environments.