In the last decade, the developments in healthcare technologies have been increasing progressively in practice. Healthcare applications such as ECG monitoring, heartbeat analysis, and blood pressure control connect with external servers in a manner called cloud computing. The emerging cloud paradigm offers different models, such as fog computing and edge computing, to enhance the performances of healthcare applications with minimum end-to-end delay in the network. However, many research challenges exist in the fog-cloud enabled network for healthcare applications. Therefore, in this paper, a Critical Healthcare Task Management (CHTM) model is proposed and implemented using an ECG dataset. We design a resource scheduling model among fog nodes at the fog level. A multi-agent system is proposed to provide the complete management of the network from the edge to the cloud. The proposed model overcomes the limitations of providing interoperability, resource sharing, scheduling, and dynamic task allocation to manage critical tasks significantly. The simulation results show that our model, in comparison with the cloud, significantly reduces the network usage by 79%, the response time by 90%, the network delay by 65%, the energy consumption by 81%, and the instance cost by 80%.