The limited availability of labeled chest X-ray datasets is a significant bottleneck in the development of medical imaging methods. Self-supervised learning (SSL) can mitigate this problem by training models on unlabeled data. Furthermore, self-supervised pretraining has yielded promising results in visual recognition of natural images but has not been given much consideration in medical image analysis. In this work, we propose a self-supervised method, DINO-CXR, which is a novel adaptation of a self-supervised method, DINO, based on a vision transformer for chest X-ray classification. A comparative analysis is performed to show the effectiveness of the proposed method for both pneumonia and COVID-19 detection. Through a quantitative analysis, it is also shown that the proposed method outperforms state-of-the-art methods in terms of accuracy and achieves comparable results in terms of AUC and F-1 score while requiring significantly less labeled data.