Clinical AI systems operating in decentralized healthcare settings must balance predictive performance with interpretability, particularly under strict regulatory and infrastructural constraints. However, most existing federated learning approaches lack symbolic reasoning capabilities, limiting their transparency, adaptability to domain shifts, and practical compliance with standards like the EU AI Act. To address this, we propose a neuro-symbolic federated learning framework that tightly couples deep neural approximators with symbolic rule engines across distributed healthcare nodes. Our architecture integrates a logic-aware aggregation mechanism, enabling interpretable policy induction and semantic preservation even under non-i.i.d. client distributions. Additionally, we introduce an ablation-guided symbolic distillation pipeline that quantifies the effect of explainability components under varying client demographics and data sparsity scenarios. The system supports adaptive reasoning across sites and complies with real-time traceability demands without compromising performance. Empirical evaluations on three cross-institutional medical datasets demonstrate a 17.3% improvement in generalization accuracy under domain shift and a 26.5% reduction in semantic divergence when logic-aware modules are enabled. The average model convergence time remained within a 5.4% deviation from standard FL baselines, indicating negligible computational overhead. Statistical validations using t-tests (p < 0.01) confirm the robustness of performance gains across unseen hospital cohorts. This work establishes a precedent for trustworthy federated AI that not only learns from distributed, heterogeneous data but also offers transparent, domain-aware decision pathways—positioning it as a viable candidate for deployment in critical, explainability-driven clinical environments.