Vision-language foundation models have emerged as powerful general-purpose representation learners with strong potential for multimodal understanding, but their deterministic embeddings often fail to provide the reliability required for high-stakes biomedical applications. This work introduces MedProbCLIP, a probabilistic vision-language learning framework for chest X-ray and radiology report representation learning and bidirectional retrieval. MedProbCLIP models image and text representations as Gaussian embeddings through a probabilistic contrastive objective that explicitly captures uncertainty and many-to-many correspondences between radiographs and clinical narratives. A variational information bottleneck mitigates overconfident predictions, while MedProbCLIP employs multi-view radiograph encoding and multi-section report encoding during training to provide fine-grained supervision for clinically aligned correspondence, yet requires only a single radiograph and a single report at inference. Evaluated on the MIMIC-CXR dataset, MedProbCLIP outperforms deterministic and probabilistic baselines, including CLIP, CXR-CLIP, and PCME++, in both retrieval and zero-shot classification. Beyond accuracy, MedProbCLIP demonstrates superior calibration, risk-coverage behavior, selective retrieval reliability, and robustness to clinically relevant corruptions, underscoring the value of probabilistic vision-language modeling for improving the trustworthiness and safety of radiology image-text retrieval systems.