Given the critical role of medical information in medical cryptographic processes, it is essential to have rigorous security validation to prevent any vulnerabilities in the authentication process that could lead to unauthorized access and access to clinical information and allow unauthorized treatment decisions to be made. This study introduces a formal security-proof framework to assess the security of authentication, confidentiality, integrity, anonymity and security of session-keys in connected healthcare settings. The proposed protocol uses finite field operations, cryptographic hashes, key derivation functions, random nonces, timestamps and medical professionals and patients. Adversarial games, mutual-authentication proofs, session-key indistinguishability and resistance to replay, impersonation, man-in-the-middle and privileged-insider attacks are discussed in the context of security. The formal reasoning approach has a negligible adversarial advantage, given the computational assumptions. Comparative evaluation shows the overhead of the authentication process is lower, but the protection of the medical devices is still high, even when considering the limited resources of medical devices. The framework provides a mathematically sound basis for designing reliable, private and effective healthcare communication systems in a heterogeneous clinical network and infrastructure.