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During the COVID-19 pandemic, engagement in various remote activities such as online education and meetings has increased. However, since the conventional online environments typically provide simple streaming services using cameras and microphones, there have limitations in terms of physical expression and experiencing real-world activities such as cultural and economic activities. Recently, metaverse environments, three-dimensional virtual reality that use avatars, have attracted increasing attention as a means to solve these problems. Thus, many metaverse platforms such as Roblox, Minecraft, and Fortnite have been emerging to provide various services to users. However, such metaverse environments are potentially vulnerable to various security threats because the users and platform servers communicate through public channels. In addition, sensitive user data such as identity, password, and biometric information are managed by each platform server. In this paper, we design a system model that can guarantee secure communication and transparently manage user identification data in metaverse environments using blockchain technology. We also propose a mutual authentication scheme using biometric information and Elliptic Curve Cryptography (ECC) to provide secure communication between users and platform servers and secure avatar interactions between avatars and avatars. To demonstrate the security of the proposed mutual authentication scheme, we perform informal security analysis, Burrows–Abadi–Needham (BAN) logic, Real-or-Random (ROR) model, and Automated Validation of Internet Security Protocols and Applications (AVISPA). In addition, we compare the computation costs, communication costs, and security features of the proposed scheme with existing schemes in similar environments. The results demonstrate that the proposed scheme has lower computation and communication costs and can provide a wider range of security features than existing schemes. Thus, our proposed scheme can be used to provide secure metaverse environments.