We propose a secure and lightweight, post-quantum authentication framework for Smart Healthcare Systems (SHS), to address the critical need for secure and efficient data management in IoT-driven healthcare environments. Leveraging the Ring Learning With Errors (RLWE) problem with the transparency of blockchain, a robust post quantum cryptographic primitive, the protocol ensures resistance against quantum attacks which threaten traditional cryptographic methods. The protocol ensures quantum-resistant, tamper-proof data sharing across hospitals, patients and regulatory bodies. A smart card-based key exchange reduces overhead on IoT devices, while support for password updates and dynamic server onboarding enhances scalability. Formally verified under the Random Oracle Model (ROM), it guarantees mutual authentication and session key agreement, with proven resilience against quantum and classical attacks, including replay and man-in-the-middle (MITM) attacks. Compliant with healthcare regulations, it ensures data privacy and auditability, paving the way for scalable, quantum-resistant SHS deployments.