Federated learning (FL) allows participants to collaboratively train machine learning models while keeping their data local, making it ideal for collaborations among healthcare institutions on sensitive data. However, in this paper, we propose a novel privacy attack called MedLeak, which allows a malicious FL server to recover high-quality site-specific private medical data from the client model updates. MedLeak works by introducing an adversarially crafted model during the FL training process. Honest clients, unaware of the insidious changes in the published models, continue to send back their updates as per the standard FL protocol. Leveraging a novel analytical method, MedLeak can efficiently recover private client data from the aggregated parameter updates, eliminating costly optimization. In addition, the scheme relies solely on the aggregated updates, thus rendering secure aggregation protocols ineffective, as they depend on the randomization of intermediate results for security while leaving the final aggregated results unaltered. We implement MedLeak on medical image datasets (MedMNIST, COVIDx CXR-4, and Kaggle Brain Tumor MRI), as well as a medical text dataset (MedAbstract). The results demonstrate that our attack achieves high recovery rates and strong quantitative scores on both image and text datasets. We also thoroughly evaluate MedLeak across different attack parameters, providing insights into key factors that influence attack performance and potential defenses. Furthermore, we demonstrate that the recovered data can support downstream tasks such as disease classification with minimal performance loss. Our findings validate the need for enhanced privacy measures in FL systems, particularly for safeguarding sensitive medical data against powerful model inversion attacks.