Nuclear Power Plants (NPPs) are physically isolated from external networks and have different operational environments than conventional information technology (IT) systems. Accordingly, NPPs were regarded as safe from external cyber-attacks. However, it was later determined that isolated networks are not safe from cyber-attacks. Malicious data injection attacks on Programmable Logic Controllers (PLCs) deployed in the safety system of NPPs are critical to nuclear facilities, as they were in the Stuxnet attack. It is necessary to monitor the integrity of PLC data and protect the PLCs from cyber threats such as modification of deployed logic or setpoints. To address this problem, this paper proposes a novel system for monitoring data integrity of PLCs using blockchain technologies. Considering the NPP environment, we developed a private blockchain system to monitor the data integrity of PLCs. The new concept that is Proof of Monitoring (PoM) for data integrity of PLCs was proposed to overcome the limitation for applying the private blockchain to the cybersecurity of NPPs. Additionally, we developed an integrity monitoring system for the Reactor Protection System (RPS)-a safety system in NPPs-using the developed blockchain. It can detect cyber-attacks (such as false code injection attacks on PLCs) and monitor which PLC integrity has been compromised in real-time. A validation experiment using a false data injection attack on PLCs was performed on the developed system, and the results confirmed that the developed system successfully monitored the modification of data in the PLCs.