Machine Learning (ML) demonstrates dermatologist-level accuracy in skin cancer diagnosis, yet its practical adoption is constrained by data silos and privacy issues. While Federated Learning (FL) addresses these limitations, it remains susceptible to data heterogeneity and gradient leakage attacks. To overcome these challenges, we introduce a privacy-preserving FL framework tailored for encrypted dermoscopic image analysis. Our proposed framework integrates a Fully Homomorphic Encryption (FHE)-enabled variant of Stochastic Controlled Averaging (SCA), enhancing model convergence with Non-IID data. To further minimize computational and communication overhead, we develop a layer-wise Packed FHE (PFHE) approach that improves the efficiency of encrypted model aggregation. Moreover, we design a lightweight, FHE-Friendly Deep Neural Network (DNN) optimized for encrypted inference. This architecture incorporates a DO-EncConv module specifically engineered to balance inference efficiency and precision within FHE computational constraints. Experimental results on the HAM10000 and ISIC2019 datasets confirm the effectiveness of our proposed framework, demonstrating F1-Score improvements of 2.2% and 4.0%, respectively, over baseline FL approaches. Additionally, our method achieves communication overhead reductions of 94.85% and 93.48%, while encrypted inference is performed in approximately 17.8 seconds per sample, with less than 2% accuracy degradation compared to centralized plaintext models. These outcomes underscore the framework's practicality and effectiveness for secure, scalable clinical deployment.