As the Internet of Things (IoT) technology is booming, the number of smart devices used in IoT has exponentially increased. These smart devices gather a massive amount of data from their surroundings for processing and communication. Handling this colossal amount of data is a challenge. Cloud computing solves this problem but with low latency and vast bandwidth consumption. However, many real-time applications, like smart transportation, automated industries, and the healthcare sector, often require efficient bandwidth utilization with low latency, which is again challenging to achieve using only cloud computing. The concept of fog computing has been transpired as a solution. Since fog devices are at the network’s edge, they typically have limited resources, making it essential to make efficient decisions about job assignments and scheduling on these devices. This paper uses a smart gateway framework for medical IoT applications, with a proposed Modified-Dynamically Prioritized Memoization-based Best Fit Algorithm (MDP-MBFA) for efficient task scheduling and resource allocation in a fog-cloud environment. The memoization technique of dynamic programming is used to efficiently allocate the urgent critical tasks of the medical IoT domain to the best fog node. Increasing the Quality of Service (QoS) by using a smart gateway at the fog layer is the primary concern of this work. Multiple QoS parameters, such as throughput, energy consumption, and system latency, are considered. The proposed framework was evaluated using the YAFS simulator and outperformed several benchmark algorithms when compared based on several QoS parameters.