Recent advances in mobile technologies and cloud computing services have inspired the development of cloud-based real-time health monitoring systems. However, the transfer of health-related data to the cloud contributes to the burden on the networking infrastructures, leading to high latency and increased power consumption. Fog computing is introduced to relieve this burden by bringing services to the users proximity. This study proposes a new fog computing architecture for health monitoring applications based on a Gigabit Passive Optical Network (GPON) access network. An Energy-Efficient Fog Computing (EEFC) model is developed using Mixed Integer Linear Programming (MILP) to optimize the number and location of fog devices at the network edge to process and analyze the health data for energy-efficient fog computing. The performance of the EEFC model at low data rates and high data rates health applications is studied. The outcome of the study reveals that a total energy saving of 36% and 52% are attained via processing and analysis the health data at the fog in comparison to conventional processing and analysis at the central cloud for low data rate application and high data rate application, respectively. We also developed a real-time heuristic; Energy Optimized Fog Computing (EOFC) heuristic, with energy consumption performance approaching the EEFC model. Furthermore, we examined the energy efficiency improvements under different scenarios of devices idle power consumption and traffic volume.