Healthcare organizations are one of the most susceptible targets to advanced cyber-attacks owing to increasing dependence on cloud computing, AI-powered clinical applications, interconnected medical devices, and EHRs. The traditional security approach is unable to tackle ransomware attacks, insider threats, identity theft, and lateral movement of hackers across networks. In our proposed work, an Agentic AI-powered Zero Trust Security Framework has been developed for healthcare organizations where agentic intelligent agents are employed for continuous user authentication, context-based risk assessment, network monitoring, device trust evaluation, and adaptation of security response in a proactive manner. In order to perform continuous verification and automated response, it utilizes ZTA, behavioral analysis, threat intelligence, reinforcement learning, and explainable AI. Agentic AI allows making decisions using the principle of dynamic least privilege, anomaly detection, and coordinated defense against threats to provide availability and compliance with regulations in the healthcare sector. Improved accuracy, response speed, and system resilience have been shown via simulations. Our architectural design can serve as a cybersecurity solution for current digital healthcare infrastructure.