The advancement of Deep Learning in medical diagnostics is often hindered by the "Data Silo" problem, where strict privacy regulations prevent the pooling of patient data across different healthcare institutions. This paper proposes a Federated Learning (FL) framework designed to train robust diagnostic models for oncology without moving sensitive patient images from their local hospital servers. We examine a decentralized architecture where only model gradients, rather than raw data, are exchanged with a central orchestrator. To further harden the system against reconstruction attacks, we integrate a Differential Privacy (DP) layer that adds calibrated noise to the local updates. The study evaluates the performance of this framework across a simulated cluster of four regional hospitals using high-resolution MRI datasets. Our results indicate that the federated model achieves a diagnostic accuracy within 2.5% of a centrally trained model while ensuring 100% compliance with data residency requirements. The findings provide a scalable roadmap for multi-institutional clinical research, allowing for the development of high-performance AI tools without compromising patient confidentiality or institutional data sovereignty.