Eye movement classification, particularly the identification of fixations and saccades, plays a vital role in advancing our understanding of neurological functions and cognitive processing. Conventional modalities of data, such as RGB webcams, often face limitations such as motion blur, latency and susceptibility to noise. Neuromorphic Vision Sensors, also known as event cameras (ECs), capture pixel-level changes asynchronously and at a high temporal resolution, making them well suited for detecting the swift transitions inherent to eye movements. However, the resulting data are sparse, which makes them less well suited for use with conventional algorithms. Spiking Neural Networks (SNNs) are gaining attention due to their discrete spatio-temporal spike mechanism ideally suited for sparse data. These networks offer a biologically inspired computational paradigm capable of modeling the temporal dynamics captured by event cameras. This study validates the use of Spiking Neural Networks (SNNs) with event cameras for efficient eye movement classification. We manually annotated the EV-Eye dataset, the largest publicly available event-based eye-tracking benchmark, into sequences of saccades and fixations, and we propose a convolutional SNN architecture operating directly on spike streams. Our model achieves an accuracy of 94% and a precision of 0.92 across annotated data from 10 users. As the first work to apply SNNs to eye movement classification using event data, we benchmark our approach against spiking baselines such as SpikingVGG and SpikingDenseNet, and additionally provide a detailed computational complexity comparison between SNN and ANN counterparts. Our results highlight the efficiency and robustness of SNNs for event-based vision tasks, with over one order of magnitude improvement in computational efficiency, with implications for fast and low-power neurocognitive diagnostic systems