Medical data exchange between diverse e-health entities can lead to a better healthcare quality, improving the response time in emergency conditions, and a more accurate control of critical medical events (e.g., national health threats or epidemics). However, exchanging large amount of information between different e-health entities is challenging in terms of security, privacy, and network loads, especially for large-scale healthcare systems. Indeed, recent solutions suffer from poor scalability, computational cost, and slow response. Thus, this article proposes medical-edge-blockchain (MEdge-Chain), a holistic framework that exploits the integration of edge computing and blockchain-based technologies to process large amounts of medical data. Specifically, the proposed framework describes a healthcare system that aims to aggregate diverse health entities in a unique national healthcare system by enabling swift, secure exchange, and storage of medical data. Moreover, we design an automated patients monitoring scheme, at the edge, which enables the remote monitoring and efficient discovery of critical medical events. Then, we integrate this scheme with a blockchain architecture to optimize medical data exchanging between diverse entities. Furthermore, we develop a blockchain-based optimization model that aims to optimize the latency and computational cost of medical data exchange between different health entities, hence providing effective and secure healthcare services. Finally, we show the effectiveness of our system in adapting to different critical events, while highlighting the benefits of the proposed intelligent health system.