6G technology is now attracting many scientific researchers due to its prominent features including high mobility, high data rate, high operating frequency, and ultra-low end-to-end delay. Compared to the 5G, 6G has certain advantages that can transform smart healthcare to a new advanced intelligent healthcare system, where multiple issues and concerns of the services (e.g., slow ambulance response) are effectively resolved. In such a communication system, patients and service providers communicate with each other via an open internet channel. Since the healthcare data is important and very sensitive, security and privacy in the healthcare network become prominent. In this paper, we introduce a 6G-aided intelligent healthcare environment. Our work also proposes a solution called Centerless User-Controlled Single Sign-On (CL-UCSSO) for achieving a convenient and cost-saving communication in the constructed multi-server system. A threefactor mechanism (combining smart card, password and biometrics) and time bound property are integrated to design the protocol with fast authentication that allows patients and providers to efficiently establish secure communications. Security proof of the proposed protocol is provided using well-known verification tools including RoR model, AVISPA simulation and BAN logic. Results of performance comparisons on various aspects show that our work provides more functionalities and incurs less cost compared with the related works.