Medical image classification plays a crucial role in computer-aided clinical diagnosis. While deep learning techniques have significantly enhanced efficiency and reduced costs, the privacy-sensitive nature of medical imaging data complicates centralized storage and model training. Furthermore, low-resource healthcare organizations face challenges related to communication overhead and efficiency due to increasing data and model scales. This paper proposes a novel privacy-preserving medical image classification framework based on federated learning to address these issues, named FedMIC. The framework enables healthcare organizations to learn from both global and local knowledge, enhancing local representation of private data despite statistical heterogeneity. It provides customized models for organizations with diverse data distributions while minimizing communication overhead and improving efficiency without compromising performance. Our FedMIC enhances robustness and practical applicability under resource-constrained conditions. We demonstrate FedMIC's effectiveness using four public medical image datasets for classical medical image classification tasks.