On the one hand, the Internet of Medical Things (IoMT) in healthcare systems has emerged as a promising technology to monitor patients’ health and provide reliable medical services, especially in remote and underserved areas. On the other hand, in disaster scenarios, the loss of communication infrastructure can make it challenging to establish reliable communication and to provide timely first aid services. To address this challenge, unmanned aerial vehicles (UAVs) have been adopted to assist hospital centers in delivering medical care to hard-to-reach areas. Despite the potential of UAVs to improve medical services in emergency scenarios, their limited resources make their security critical. Therefore, developing secure and efficient communication protocols for IoMT networks using UAVs is a vital research area that can help ensure reliable and timely medical services. In this paper, we introduce a novel Stackelberg security-based game theory algorithm, named Stackelberg ad hoc on-demand distance vector (SBAODV), to detect and recover data affected by black hole attacks in IoMT networks using UAVs. Our proposed scheme utilizes the substantial Stackelberg equilibrium (SSE) to formulate strategies that protect the system against attacks. We evaluate the performance of our proposed SBAODV scheme and compare it with existing routing schemes. Our results demonstrate that our proposed scheme outperforms existing schemes regarding packet delivery ratio (PDR), networking load, throughput, detection ratio, and end-to-end delay. Specifically, our proposed SBAODV protocol achieves a PDR of 97%, throughput ranging from 77.7 kbps to 87.3 kbps, and up to 95% malicious detection rate at the highest number of nodes. Furthermore, our proposed SBADOV scheme offers significantly lower networking load (7% to 30%) and end-to-end delay (up to 30%) compared to existing routing schemes. These results demonstrate the efficiency and effectiveness of our proposed scheme in ensuring reliable and secure communication in IoMT emergency scenarios using UAVs.