To effectively treat patients, health care providers must be able to detect diseases early and diagnose them accurately. Deep learning and computer vision have recently enhanced the diagnostic accuracy of skin cancer through image-classification models. However, the centralized learning (CL) method is problematic in terms of data privacy because of the constraint of transferring substantial volumes of data to a central server. In this study, we investigate edge intelligence training in a multi-institutional edge environment to address data privacy, machine learning training delay, and training data limitations at each healthcare center. We considered skin cancer image classification as our use case in light of skin cancer diagnosis and assisting health care experts. Initially, we focused on the training delays induced by communication and computational latency. The edge-average federated learning (Edge-Avg) method improved the training latency by 24% and 47% compared with the federated learning (Fed-Avg) and CL approaches, respectively. This indicates reduced processing time, which is particularly crucial in the field of medical diagnostics. In terms of accuracy, our Edge-Avg approach surpassed the Fed-Avg method by achieving a better accuracy of 93.94% compared to the traditional method in the classification of skin cancer images into benign or malignant categories. This study provides a low-latency and accurate solution to assist medical professionals in assessing the status of suspected skin cancer cases.