Ophthalmic surgical robots offer superior stability and precision by reducing the natural hand tremors of human surgeons, enabling delicate operations in confined surgical spaces. Despite the advancements in developing vision- and force-based control methods for surgical robots, preoperative navigation remains heavily reliant on manual operation, limiting the consistency and increasing the uncertainty. Existing eye gaze estimation techniques in the surgery, whether traditional or deep learning-based, face challenges including dependence on additional sensors, occlusion issues in surgical environments, and the requirement for facial detection. To address these limitations, this study proposes an innovative eye localization and tracking method that combines machine learning with traditional algorithms, eliminating the requirements of landmarks and maintaining stable iris detection and gaze estimation under varying lighting and shadow conditions. Extensive real-world experiment results show that our proposed method has an average estimation error of 0.58 degrees for eye orientation estimation and 2.08-degree average control error for the robotic arm's movement based on the calculated orientation.