Deep learning models have revolutionized medical imaging and diagnostics, yet their opaque nature poses challenges for clinical adoption and trust. Amongst approaches to improve model interpretability, concept-based explanations aim to provide concise and human-understandable explanations of any arbitrary classifier. However, such methods usually require a large amount of manually collected data with concept annotation, which is often scarce in the medical domain. In this paper, we propose Conceptual Counterfactual Explanations for Chest X-ray (CoCoX), which leverages the joint embedding space of an existing vision-language model (VLM) to explain black-box classifier outcomes without the need for annotated datasets. Specifically, we utilize textual concepts derived from chest radiography reports and a pre-trained chest radiography-based VLM to explain three common cardiothoracic pathologies. We demonstrate that the explanations generated by our method are semantically meaningful and faithful to underlying pathologies.