The Internet of Medical Things (IoMT) is a unification of smart healthcare devices, tools, and software, which connect various patients and other users to the healthcare information system through the networking technology. It further reduces unnecessary hospital visits and the burden on healthcare systems by connecting the patients to their healthcare experts (i.e., doctors) and allows secure transmission of healthcare data over an insecure channel (e.g., the Internet). Since Artificial Intelligence (AI) has a great impact on the performance and usability of an information system, it is important to include its modules in a healthcare information system, which will be very helpful for the prediction of some phenomena, such as chances of getting a heart attack and possibility of a tumor, from the collected and analysed healthcare data. To mitigate these issues, in this paper, a new AI-enabled lightweight, secure communication scheme for an IoMT environment has been designed and named as ASCP-IoMT, in short. The security analysis of ASCP-IoMT is performed in different ways, such as an informal way and a formal way (through the random oracle model). ASCP-IoMT performs better than other similar schemes and provides superior security with extra functionality features as compared those for the existing state of art solutions. A practical implementation of ASCP-IoMT is also performed in order to measure its impact on various network performance parameters. The end to end delay values of ASCP-IoMT are 0.01587, 0.07440 and 0.17097 seconds and the throughput values of ASCP-IoMT are 5.05, 10.88 and 16.41 bps under the different considered cases, respectively. For AI-based Big data analytics phase, the values of computation time (seconds) for decision tree, support vector machine (SVM), and logistic regression are measured as 0.19, 0.23, and 0.27, respectively. Moreover, the different values of accuracy for decision tree, SVM and logistic regression are 84.24%, 87.57%, and 85.20%, respectively. From these values, it is clear that decision tree method requires less time than the other considered techniques, whereas accuracy is high in case of SVM.