The increasing use of medical applications using mobile devices has led to increased concerns about data security. Mobile Edge Computing (MEC) is an emerging technology that can be used to improve end-user services, especially in medical applications that require low latency and high bandwidth. MEC provides a promising solution that brings data processing closer to the end user to process requests faster. In doing so, security concerns about communication links and MEC server faults become a problem for MEC architectures and the concerned applications. In this paper, we propose a secure and fault-tolerant mechanism for the Internet of Medical Things (IoMT) based on an MEC architecture in a Software-Defined Network. Then, we present a secure scheme that consists of four main steps. (1) Establishment of secure virtual private network connection to secure network communication, (2) monitoring of network traffic to detect threats, (3) application of intrusion prevention measures, and (4) updating of security rules to prevent future attacks. The fault-tolerant aspect is developed based on the Kubernetes cluster to avoid service disruption in case of faults on a given MEC server. The simulation results depict substantial improvements, including reduced latency, better workload equilibrium, increased achievable throughput, and reduced retransmissions between servers. Our proposed scheme is effective in detecting distributed denial of service attacks. Moreover, in the event of an MEC server fault, the service is not interrupted and the architecture continues to render optimal services.