The security and privacy of electronic health records (EHRs) have received considerable attention from healthcare workers and researchers. To ensure security, various encryption and decryption schemes as well as key management protocols have been developed. However, owing to sharing and scalability issues, additional security technologies have been proposed. Nonetheless, these technologies cause other problems, such as efficiency issues. Blockchain-based EHR management systems have been proposed to overcome computational overhead. However, because most blockchain systems are installed by outsourcing companies, EHRs may be leaked to the company. Hence, we herein propose a blockchain-based EHR management scheme with proxy re-encryption. In this scheme, we set a proxy server that re-encrypts the ciphertext between file servers, thereby solving EHR sharing issues. Furthermore, because the server is separated from the blockchain system, the outsourcing company cannot manipulate the server or access the records. In addition, the blockchain assists in access control by using smart contracts, thereby enabling secure and efficient EHR sharing. By performing security analysis, we prove that our proposed scheme solves the aforementioned security problems. In addition, we experimentally demonstrate the efficient operation of the proposed system.