With the growth and popularity of the utilization of medical images in smart healthcare, the security of these images using watermarks is one of the most recent research topics. This algorithm is based on the joint use of dual watermarking, nature-inspired optimization, and encryption schemes utilizing redundant-discrete wavelet transform (RDWT) and randomized-singular value decomposition (RSVD). The key idea of the proposed method is to embed system encoded media access control (MAC) address in patient's ID card image via discrete wavelet transform (DWT) to generate the final mark. Afterward, embed the generated watermark into computed tomography (CT) scan images of the COVID-19 patient and general images through employing the RDWT and RSVD. Further, we use a hybrid of particle swarm optimization (PSO) and Firefly optimization techniques to determine the optimal scaling factor for embedding purposes. After that, the watermarked CT scan image is encrypted using an encryption technique based on a nonlinear-chaotic map, random permutation, and singular value decomposition (SVD). Extensive evaluations establish the benefit of our proposed algorithm over the traditional schemes. The optimal robustness is more effective than the five traditional schemes at lower computational efficiency.