In this work, a new quaternion-based method for color image watermarking is proposed. In this method, a novel set of quaternion radial substituted Chebyshev moments (QRSCMs) is presented for robust geometrically invariant image watermarking. An efficient computational method is proposed for highly accurate, fast, and numerically stable QRSCMs in polar coordinates. The proposed watermarking method consists of three stages. In the first stage, the Arnold transform is used to improve the security of the watermarking scheme by scrambling the binary watermark. In the second stage, the proposed accurate and stable QRSCMs of the host color image are computed. In the third stage, the encrypted binary watermark is embedded into the host image by employing the quantization technique on selected-magnitude QRSCMs where the watermarked color image is obtained by adding the original host color image to the compensation image. Then, the binary watermark can be extracted directly without using the original image from the magnitudes of QRSCMs. Numerical experiments are performed where the performance of proposed method is compared with the existing quaternion moment-based watermarking methods. The comparison clearly shows that the proposed method is very efficient in terms of the visual imperceptibility capability and the robustness under different attacks compared to the existing quaternion moment-based watermarking algorithms.