Amblyopia is a neurodevelopmental visual disorder that affects approximately 3-5% of children globally and it can lead to vision loss if it is not diagnosed and treated early. Traditional diagnostic methods, which rely on subjective assessments and expert interpretation of eye movement recordings presents challenges in resource-limited eye care centers. This study introduces a new approach that integrates the Gemini large language model (LLM) with eye-tracking data to develop a classification tool for diagnosis of patients with amblyopia. The study demonstrates: (1) LLMs can be successfully applied to the analysis of fixation eye movement data to diagnose patients with amblyopia; and (2) Input of medical subject matter expertise, introduced in this study in the form of medical expert augmented generation (MEAG), is an effective adaption of the generic retrieval augmented generation (RAG) approach for medical applications using LLMs. This study introduces a new multi-view prompting framework for ophthalmology applications that incorporates fine granularity feedback from pediatric ophthalmologist together with in-context learning to report an accuracy of 80% in diagnosing patients with amblyopia. In addition to the binary classification task, the classification tool is generalizable to specific subpopulations of amblyopic patients based on severity of amblyopia, type of amblyopia, and with or without nystagmus. The model reports an accuracy of: (1) 83% in classifying patients with moderate or severe amblyopia, (2) 81% in classifying patients with mild or treated amblyopia; and (3) 85% accuracy in classifying patients with nystagmus. To the best of our knowledge, this is the first study that defines a multi-view prompting framework with MEAG to analyze eye tracking data for the diagnosis of amblyopic patients.