As COVID-19 continues to spread around the world, the healthcare industry has accelerated the transformation to digital healthcare services. In the era of big data, many hospitals prefer to use remote cloud servers to store and manage massive electronic medical data. However, cloud-assisted medical data systems cannot guarantee the confidentiality, integrity, and availability of data. Searchable encryption can effectively address the above challenges by enabling data search on the ciphertext, which achieves the availability of medical data while ensuring data security and privacy. However, the search server may return mismatched search results due to economic interests or single points of failure. Blockchain is a decentralized computing paradigm with public verifiability, which provides an efficient solution to this problem. However, the existing blockchain-based searchable encryption solutions do not consider the flexible search function of multiple users and the restriction of encrypted data for medical scenarios. Therefore, we propose a blockchain-based multiuser normalized searchable encryption (BNSE) scheme and design a blockchain-based normalized searchable encryption system for medical data (BNSEM) based on the scheme. To verify the practicality of the system, we evaluate the performance from both theoretical and experimental aspects.