Medical image analysis using deep learning requires large and diverse datasets that are often siloed within a Healthcare Institution (HI) due to privacy concerns and regulatory restrictions. With sensitive medical data, traditional training approaches usually require data centralization. This raises substantial concerns, including privacy and security. As a solution, we propose Fed-Eye, a Federated Learning (FL) based framework specifically designed for ophthalmological disease classification across multiple HI while enabling privacy-preserving and addressing the problems of domain shift and data heterogeneity. We evaluate Fed-Eye across diverse ophthalmological datasets spanning diabetic retinopathy, glaucoma, and other mixed pathologies. Our method attains competitive performance with state-of-the-art. This high performance is achieved while keeping the privacy of data. This indicates the potential of our method for developing privacy-preserving and clinically valuable AI systems in ophthalmology.