The continuous evolution of digital medical imaging underscores the critical importance of ensuring image security. Conventional watermarking techniques for robustness typically embed copyright details into the image, thereby altering its original form and making them unsuitable for domains such as the military and medicine, where absolute image fidelity is essential. To address this limitation, this paper proposes a robust zero-watermarking algorithm that enables copyright protection of medical images without modifying the host image. In this approach, fractional-order polar harmonic Fourier moments (FrPHFMs) were employed as invariant geometric descriptors in the construction of zero-watermarks. First, FrPHFMs were computed, and then the most accurate descriptors were selected to provide improved performance and strong geometric invariance. The extracted features are then integrated with a binary watermark using a 1D chaotic map (1D-RSS), formulated from reciprocal and squared sine functions and incorporating an exclusive or (XOR) operation, to construct the zero-watermark. Building on this design, a hybrid FrPHFM-chaos zero-watermarking algorithm was introduced, in which FrPHFMs ensure strong resilience to geometric distortions, while the chaotic map enhances security through its sensitivity to initial parameters. Experimental results demonstrated that the proposed algorithm exhibits high resistance to geometric distortions, conventional attacks, and combined attacks, outperforming existing methods. It achieves bit error rate (BER) < 0.001 and normalized correlation (NC) > 0.9, confirming its superior robustness.