Cloud computing is vital in various applications, such as healthcare, transportation, governance, and mobile computing. When using a public cloud server, it is mandatory to be secured from all known threats because a minor attacker's disturbance severely threatens the whole system. A public cloud server is posed with numerous threats; an adversary can easily enter the server to access sensitive information, especially for the healthcare industry, which offers services to patients, researchers, labs, and hospitals in a flexible way with minimal operational costs. It is challenging to make it a reliable system and ensure the privacy and security of a cloud-enabled healthcare system. In this regard, numerous security mechanisms have been proposed in past decades. These protocols either suffer from replay attacks, are completed in three to four round trips or have maximum computation, which means the security doesn't balance with performance. Thus, this work uses a fuzzy extractor method to propose a robust security method for a cloud-enabled healthcare system based on Elliptic Curve Cryptography (ECC). The proposed scheme's security analysis has been examined formally with BAN logic, ROM and ProVerif and informally using pragmatic illustration and different attacks' discussions. The proposed security mechanism is analyzed in terms of communication and computation costs. Upon comparing the proposed protocol with prior work, it has been demonstrated that our scheme is 33.91% better in communication costs and 35.39% superior to its competitors in computation costs.