Presently, blockchain technology is used to secure electronic medical records (EMR) and an arrangement of multiple servers as off-chain storage is advocated to minimize the storage overhead of the medical blockchain. Therefore, authorized access to the medical records stored in multiple servers, needs a secure multi-server-based authentication system. However, existing blockchain-based systems for medical data storage do not consider an authentication system for a multi-server environment between patients and multiple medical servers. In this paper, a blockchain of patients' healthcare data using off-chain storage is considered to ensure scalability of the blockchain. The blockchain contains the hash value of the medical data while multiple servers are used as off-chain storage for storing the original data. A patient can access those servers in a single enrollment under a multi-server authentication system and can share his/her health data with an authorized care provider. Formal as well as informal security analysis is used to check the security of the proposed protocol. The formal security analysis like AVISPA tool is used to check the replay attacks and the BAN logic is applied to check the mutual authentication of the proposed protocol. At the same time, a rigorous informal security analysis confirms that our scheme is secured against various known attacks. We have compared our scheme with recent research with respect to security functionality, computation cost, and communication cost.