In this study, we examine the Hyers–Ulam stability of a fractal–fractional mathematical model for the transmission dynamics of the Zika virus. The model is formulated using the Atangana–Baleanu fractional operator with a Mittag–Leffler kernel, thereby incorporating both memory effects and fractal properties commonly observed in biological systems. This approach provides a more realistic representation of disease spread than classical integer-order models. First, the existence and uniqueness of solutions are established to ensure the well-posedness of the proposed system. Subsequently, fixed-point theory is applied to derive sufficient conditions for Hyers–Ulam stability, demonstrating that the model remains stable under small perturbations. This stability property is essential for confirming the model's reliability and robustness in practical applications. In addition, numerical simulations are performed to support theoretical findings and illustrate the qualitative behavior of the Zika virus transmission process. The obtained results provide valuable insights into the influence of memory and fractal effects on epidemic dynamics. In general, the study highlights the effectiveness of the fractal–fractional framework in capturing complex biological phenomena and confirms its potential for future research in mathematical epidemiology. The proposed methodology can also be extended to analyze other infectious diseases, contributing to the development of more accurate and reliable predictive models.