It is crucial to ensure the privacy and authenticity of patients’ medical data in the medical insurance claim process, but in the current medical insurance claim process, there are some problems such as low efficiency, complex service, unreliable data and data leakage. Therefore, considering the privacy and sensitivity of patients’ medical data, we can improve the current issues by employing blockchain, smart contracts and zero-knowledge proof technology. In this paper, we propose a novel medical insurance claim scheme based on smart contracts, blockchain and zero-knowledge proof. Our scheme mainly involves two scenarios: medical insurance purchasing and medical insurance claiming. In the privacy-preserving transaction phases of the two scenarios, we can ensure the legitimacy and privacy of the transactions between the patients and the insurance companies by using a non-interactive zero-knowledge proof and the homomorphic encryption algorithm under the Decisional Bilinear Diffie–Hellman (DBDH) assumption. In the identity privacy-preserving phases of the two scenarios, we can ensure the legitimacy and the privacy of patients’ identities by integrating Schnorr protocol and Fiat–Shamir heuristic method. The security analysis, the computation cost and the communication cost of our scheme are given. Compared with our referred schemes, the performance evaluation shows that our scheme not only meets the requirements of the legality of the medical insurance claim, but also ensures the authenticity and privacy of the patients’ medical data. Moreover, the experimental results demonstrate that our scheme is feasible and has an acceptable time overhead.