Glaucoma is a leading cause of irreversible blindness worldwide, often progressing undiagnosed due to asymptomatic early stages and limited access to specialist care. To address these barriers, we present a feasibility study of Glaucoma-PAIR (Privacy-preserving AI-based Referral), a computer-aided system developed using federated learning (FL). The study was conducted across a multi-centric network of three heterogeneous clinical sites in Portugal, two tertiary large public hospitals and one private clinic, each with distinct patient demographics, imaging equipment, and data distributions. The system leverages color fundus photography and expert-labeled cases to train a glaucoma classification model, without transferring sensitive patient data across institutions by employing FL, ensuring compliance with institutional governance and data protection regulations. Our work addresses major challenges in clinical AI, including privacy, generalizability, and integration into real-world workflows. Through close collaboration with ophthalmologists, we identified key constraints in existing referral pathways and incorporated those insights into the study design. Notably, the federated global model achieved performance comparable to a centralized model trained on pooled data, improved the average sensitivity which is a critical metric for a screening tool, and showed significant performance gains at the most clinically diverse site. This study provides a practical demonstration of responsible machine learning, combining privacy-preserving operations with clinical feasibility. Our findings highlight the potential of federated learning to enable the development of scalable and equitable AI tools, to support patient triage for a glaucoma specialist, particularly in settings with limited ophthalmology accessibility, promoting access to earlier diagnosis and care. We discuss the implications for future deployment and integration into national screening workflows.