Digital Watermarking has been widely employed for content authentication by securely embedding patient private information within medical images. In this work, we present a novel watermarking, which employs hybrid Multiscale/Multiresolution frequency coefficients selection using the Fast Discrete Curvelet Transform (FDCT) in conjunction with Singular Value Decomposition (SVD). In order to add an extra layer of security, the Radon Transform (RT) is applied on the watermarks before embedding for the sake of robustness and security due to RT properties of capturing image directionality which is essential against various attacks. The proposed method attained promising results and has shown that the imperceptibility of watermarked medical images is higher than 55 dB for all medical images. In addition, the scheme has shown an improved robustness compared with existing related work.