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Critical healthcare application tasks require a real-time response because it affects patients' life. Fog computing is the best solution to get a fast response and less energy consumption in healthcare. However, current solutions face difficulties in scheduling the tasks to the correct computing devices based on their priorities and capacity to meet the tasks' deadlines and resource limitations with minimal latency. Furthermore, challenges of load balancing and prioritization are raised when dealing with inadequate computing resources and telecommunication networks while obtaining the best scheduling of emergency healthcare tasks. In this study, a fog computing resource management (FRM) model is proposed, which the proposed model has three main solutions. Firstly, resource availability is calculated according to the average execution time of each task. Secondly, load balancing is enhanced by proposing a hybrid approach that combines the multi-agent load balancing algorithm and the throttled load balancing algorithm. Thirdly, task scheduling is done based on priority, resource availability, and load balancing. The results have been acquired using the iFogSim toolkit. Two datasets are used in this study, the blood pressure dataset was acquired from the UTeM clinic, and the ECG dataset was acquired from the University of California at Irvine. Both datasets are integrated to enlarge the attributes and get accurate results. The results demonstrate the effectiveness of managing resources and optimizing task scheduling and balancing in a fog computing environment. In comparison with other research studies, the FRM model outperforms delay by 55%, response time by 72%, cost by 72%, and energy consumption by 70%. [ABSTRACT FROM AUTHOR]