Medical images (e.g. X-rays) are often acquired at high resolutions with large dimensions in order to capture fine-grained details. In this work, we address the challenge of compressing medical images while preserving fine-grained features needed for diagnosis, a property known as diagnostic losslessness. To this end, we (1) use over one million medical images to train a domain-specific neural compressor and (2) develop a comprehensive evaluation suite for measuring compressed image quality. Extensive experiments demonstrate that large-scale, domain-specific training of neural compressors improves the diagnostic losslessness of compressed images when compared to prior approaches.