Continuous monitoring of neonatal behavior in the Neonatal Intensive Care Unit (NICU) is essential for early detection of neurological disorders. Among behavioral indicators, eye state (open vs. closed) serves as a clinically relevant marker for alertness, sedation, and responsiveness. This study presents a deep learning-based system for automated eye state detection in NICU video recordings. Using a manually labeled dataset of 7,388 facial frames extracted from 154 clinical videos, we trained and evaluated binary classifiers based on VGG16 and VGG19 convolutional neural network architectures. A five-fold cross-validation scheme was implemented to ensure subject-independent evaluation. The models achieved mean frame-level accuracies above 0.86 and AUC-PR values of 0.97. Additionally, video-level evaluation under realistic conditions yielded up to 0.79 accuracy and 0.84 AUC-PR. These results support the feasibility of integrating eye state detection into broader AI frameworks for neonatal monitoring and early neurological assessment.