Remote Patient Monitoring (RPM) is a form of telehealth or virtual health that strengthens online medical services and allows delivering healthcare remotely. Nowadays, remote patient monitoring systems (RPMS) are widely used by healthcare providers to remotely monitor the vital signs of patients. As the RPM field expands, concerns about efficient and secure medical data transmission are raised. This kind of patient medical data is collected by the mean of Internet of Healthcare Things (IoHT) or sometimes denoted as Internet of Medical Things (IoMT) devices. The collected data needs to be stored and retrieved with highly assured levels of security and privacy as the data comprises private and critical patients’ information. To secure medical data, this paper proposes a blockchain-based architecture to manage access control to medical data and to preserve patient’s data privacy. The blockchain-based system is built on Hyperledger Fabric, a permissioned distributed ledger solution, and the ledgers and transactions are stored in the cloud. The proposed architecture is designed to contribute to the robustness of the RPM systems and to avoid recorded security limitations in commonly used permissioned blockchains methods. Performance evaluation has proved the robustness and superiority of the proposed system in terms of data confidentiality, integrity, availability, traceability, scalability, and data privacy while integrated with the RPM services.