The Internet of Medical Things connects patient sensors to hospitals, offering real-time health monitoring and early alarm detection. While this improves healthcare, it also raises concerns about data privacy, secure sharing, and reliable learning across multiple hospitals. To solve these issues, we design a federated learning system that uses Elliptic Curve Cryptography for lightweight encryption, Recurrent Neural Networks for patient alarm prediction, and a blockchain layer based on Proof of Stake deployed on the Sepolia network. In this system, hospitals encrypt sensor data and send model updates instead of raw data, protecting patient privacy. The central server trains a global model using federated learning, and blockchain records all communications to ensure transparency and prevent tampering. Experiments across ten hospitals show improved accuracy, reduced communication cost, and secure model sharing. This framework demonstrates a practical way to combine AI, cryptography, and blockchain for secure healthcare.