Sustainable smart healthcare applications are those in which health services can be provided to remotely located patients through the Internet without placing extra burden on environmental resources. They should be operated with minimum power consumption using biodegradable, recyclable, and environmentally friendly healthcare equipment and products. In an Internet of Medical Things (IoMT)-enabled sustainable smart healthcare environment, all the health services are capable of producing informative data whenever some raw information is provided as the input or are capable of performing work on their own with less intervention from humans. As a result, they provide great advantages over the traditional healthcare system. As sustainable smart healthcare devices are operated through the Internet, it is possible that they could be attacked by various hackers. To mitigate these issues, in this paper, we propose a new access control along with a key-establishment mechanism for a sustainable smart healthcare system. The results of the security analysis showed that the proposed scheme was highly robust against a variety of passive and active attacks. In comparison to existing competing schemes, the proposed scheme is lightweight, as well as delivers high security and additional functionality. Finally, a practical demonstration of the proposed scheme is provided to show its impact on the key network performance parameters.