IoT has enabled the rapid growth of smart remote healthcare applications. These IoT-based remote healthcare applications deliver fast and preventive medical services to patients at risk or with chronic diseases. However, ensuring data security and patient privacy while exchanging sensitive medical data among medical IoT devices is still a significant concern in remote healthcare applications. Altered or corrupted medical data may cause wrong treatment and create grave health issues for patients. Moreover, current remote medical applications' efficiency and response time need to be addressed and improved. Considering the need for secure and efficient patient care, this paper proposes a lightweight Blockchain-based and Fog-enabled remote patient monitoring system that provides a high level of security and efficient response time. Simulation results and security analysis show that the proposed lightweight blockchain architecture fits the resource-constrained IoT devices well and is secure against attacks. Moreover, the augmentation of Fog computing improved the responsiveness of the remote patient monitoring system by 40%.