Robust watermarking is a valuable methodology used in protecting the copyright and securing digital images. In this paper, new fractional-order multi-channel orthogonal exponent moments (MFrEMs) and their invariants to geometric transformations are derived for the first time. We utilized these highly accurate moments to construct a new robust watermarking algorithm for color images. This algorithm consists of three phases. First, the bits of the binary watermark scrambled by using a 1D Sine chaotic map. Second, the fractional-order MFrEMs are calculated from the host color image. Finally, a quantization process is performed, where the scrambled bits of the binary watermark embedded into the host color image. Various experiments were conducted to test the proposed watermarking algorithm and compare it with the existing robust watermarking algorithms for color images. The obtained results ensure the proposed robust watermarking algorithm's superiority over existing algorithms regarding the visual imperceptibility and robustness against various attacks.