Fog–cloud-based hierarchical task-scheduling methods are embracing significant challenges to support e-Health applications due to the large number of users, high task diversity, and harsher service-level requirements. Addressing the challenges of fog–cloud integration, this paper proposes a new service/network-aware fog–cloud hierarchical resource-mapping scheme, which achieves optimized resource utilization efficiency and minimized latency for service-level critical tasks in e-Health applications. Concretely, we develop a service/network-aware task classification algorithm. We adopt support vector machine as a backbone with fast computational speed to support real-time task scheduling, and we develop a new kernel, fusing convolution, cross-correlation, and auto-correlation, to gain enhanced specificity and sensitivity. Based on task classification, we propose task priority assignment and resource-mapping algorithms, which aim to achieve minimized overall latency for critical tasks and improve resource utilization efficiency. Simulation results showcase that the proposed algorithm is able to achieve average execution times for critical/non-critical tasks of 0.23/0.50 ms in diverse networking setups, which surpass the benchmark scheme by 73.88%/52.01%, respectively.