The authenticity and integrity of medical images in telemedicine has to be protected. Robust reversible watermarking (RRW) algorithms provide copyright protection and the original images can be recovered at the receiver’s end. However, the existing algorithms have limitations in their ability to balance the tradeoff among robustness, imperceptibility, and embedded capacity. Some of them are even not completely reversible. Besides, most medical image watermarking algorithms are not designed for color images. To improve their performance in protecting medical color image information, we propose a novel RRW scheme based on the discrete wavelet transform (DWT). First, the DWT provides a robust solution. Second, the modification of the wavelet domain coefficient guarantees the changes of integer values in the spatial domain and ensures the reversibility of the watermarking scheme. Third, the embedding scheme makes full use of the characteristics of the original image and watermarking. This reduces the modification of the original image and ensures better imperceptibility. Lastly, the selection of the Zernike moments order for geometric correction is optimized to predict attack parameters more accurately by using less information. This enhances the robustness of the proposed scheme against geometric attacks such as rotation and scaling. The proposed scheme is robust against common and geometric attacks and has a high embedding capacity without obvious distortion of the image. The paper contributes towards improving the security of medical images in remote healthcare.