In the field of Computer Science, Artificial Intelligence and Distributed Systems, collaborative clinical AI model training requires strict patient confidentiality guarantees and minimal inter-institutional network overhead. This empirical investigation systematically examines Differentially Private Federated Learning with Adaptive Gradient Sparsification for Multi-Hospital Diagnostics through a multi-stage experimental methodology and rigorous quantitative analytical framework. Utilizing Renyi differential privacy coupled with top-k gradient compression across eight simulated clinical client hospital nodes, data were gathered across multiple operational cycles and validated against established international benchmarks. The statistical and computational results reveal that the framework maintained high diagnostic AUC (0.942) with strict differential privacy (epsilon = 1.8) while cutting network payload by 84%. Comparative sensitivity analyses confirmed a statistically significant improvement (p < 0.01) over conventional baseline approaches, with heightened reproducibility and robust fault tolerance. These comprehensive findings provide actionable theoretical insights and practical implementation guidelines for clinical AI consortia, medical imaging researchers, and healthcare privacy compliance officers. Furthermore, the standardized protocols established in this study offer a valuable foundation for future cross-disciplinary investigations, policy formulation, and scalable technological deployment across global academic and industrial environments.