Retinopathy of prematurity (ROP) is a disease that affects premature infants, where abnormal growth of the retinal blood vessels can lead to blindness unless treated accordingly. Infants considered at risk of severe ROP are monitored for symptoms of plus disease, characterized by arterial tortuosity and venous dilation at the posterior pole, with a standard photographic definition. Disagreement among ROP experts in diagnosing plus disease has driven the development of computer-based methods that classify images based on hand-crafted features extracted from the vasculature. However, most of these approaches are semi-Automated, which are time-consuming and subject to variability. In contrast, deep learning is a fully automated approach that has shown great promise in a wide variety of domains, including medical genetics, informatics and imaging. Convolutional neural networks (CNNs) are deep networks which learn rich representations of disease features that are highly robust to variations in acquisition and image quality. In this study, we utilized a U-Net architecture to perform vessel segmentation and then a GoogLeNet to perform disease classification. The classifier was trained on 3,000 retinal images and validated on an independent test set of patients with different observed progressions and treatments. We show that our fully automated algorithm can be used to monitor the progression of plus disease over multiple patient visits with results that are consistent with the experts' consensus diagnosis. Future work will aim to further validate the method on larger cohorts of patients to assess its applicability within the clinic as a treatment monitoring tool.