Advancements in machine learning (ML) have significantly revolutionized medical image analysis, prompting hospitals to rely on external ML services. However, the exchange of sensitive patient data, such as chest X-rays, poses inherent privacy risks when shared with third parties. Addressing this concern, we propose MedBlindTuner, a privacy-preserving framework leveraging fully homomorphic encryption (FHE) and a data-efficient image transformer (DEiT). MedBlindTuner enables the training of ML models exclusively on FHE-encrypted medical images. Our experimental evaluation demonstrates that MedBlindTuner achieves comparable accuracy to models trained on non-encrypted images, offering a secure solution for outsourcing ML computations while preserving patient data privacy. To the best of our knowledge, this is the first work that uses data-efficient image transformers and fully homomorphic encryption in this domain.