Skin cancer is one of the most common cancers worldwide, where early detection significantly reduces mortality rates. Integrating Internet of Things (IoT) technologies into healthcare enables continuous monitoring and early diagnosis through connected devices, offering new prospects for real-time medical assessment. Advanced machine learning classifiers have shown superior performance over human experts in diagnosing pigmented skin lesions, underscoring their potential for IoT-based applications. However, training high-precision computer-aided diagnosis (CAD) systems requires extensive data, and secure data sharing poses significant challenges due to privacy concerns in IoT healthcare networks. This paper introduces an IoT-oriented skin cancer CAD system combining federated learning and homomorphic encryption to address these issues. Federated learning supports collaborative model training across distributed IoT devices, mitigating data scarcity without compromising data privacy. Homomorphic encryption ensures that patient data remains encrypted during diagnosis, enhancing security within IoT frameworks. To achieve an efficient and high-accuracy model suited for IoT, we implemented Self-Learnable Activation Functions (SLAF), optimized for homomorphic encryption and resource constraints of IoT devices. Our system, validated using the HAM10000 dataset, achieved 94.39% accuracy with dual privacy protection, enhancing both diagnostic precision and data security. Comprehensive evaluations and comparisons with state-of-the-art frameworks confirmed the effectiveness of our approach. Results demonstrate the strong potential and practicality of our IoT-based solution for real-world healthcare, providing a secure, efficient, and accurate diagnostic tool that supports the wider adoption of IoT in medical applications.