Edge computing is a fast-growing and much needed technology in healthcare. The problem of implementing artificial intelligence on edge devices is the complexity and high resource intensity of the most known neural network data analysis methods and algorithms. The difficulty of implementing these methods on low-power microcontrollers with small memory size calls for the development of new effective algorithms for neural networks. This study presents a new method for analyzing medical data based on the LogNNet neural network, which uses chaotic mappings to transform input information. The method effectively solves classification problems and calculates risk factors for the presence of a disease in a patient according to a set of medical health indicators. The efficiency of LogNNet in assessing perinatal risk is illustrated on cardiotocogram data obtained from the UC Irvine machine learning repository. The classification accuracy reaches ~91% with the~3–10 kB of RAM used on the Arduino microcontroller. Using the LogNNet network trained on a publicly available database of the Israeli Ministry of Health, a service concept for COVID-19 express testing is provided. A classification accuracy of ~95% is achieved, and~0.6 kB of RAM is used. In all examples, the model is tested using standard classification quality metrics: precision, recall, and F1-measure. The LogNNet architecture allows the implementation of artificial intelligence on medical peripherals of the Internet of Things with low RAM resources and can be used in clinical decision support systems.