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Hospitals are rapidly adopting artificial intelligence for triage, imaging, scheduling etc., yet most deployments remain isolated point solutions locked inside departmental silos, resulting in duplicated effort, hidden risks, and unrealized enterprise value. Despite explosive growth of AI in healthcare market and accelerating investment, an estimated 70-80% of healthcare AI pilots fail to scale, largely due to governance gaps, fragmented data, and missing integration blueprints. This research proposes a hospital-specific, compliance-first, Agentic AI architecture with multiple interoperable layers, extending existing hospital AI platform models with: (i) an Agent Orchestration Layer for multi-agent workflows across clinical, operational, and financial domains, (ii) a Compliance and Policy Layer that centralizes policy-as-code for HIPAA, GDPR, the EU AI Act, DISHA Act, India's DPDP Act, and ISO/IEC security and safety standards, and (iii) a Privacy-Preserving Data Fabric that plugs federated learning, differential privacy, and secure enclaves into real-world Hospital Information Management System (HIMS) flows. Using a synthetic but structurally realistic hospital dataset and an open, ready-to-deploy prototype implementation, this study demonstrates the end-to-end orchestration of triage risk prediction, workflow optimization, and compliance logging, achieving substantial simulated reductions in task turnaround times and manual documentation effort while maintaining policy-guarded data access. The resulting architecture offers hospital leaders a pragmatic blueprint to move from ad hoc tools to a governed, globally compliant, ROI-focused AI platform that can be tailored to on-premise, hybrid and cloud-native deployments.