This work considers a numerical scheme for solving fractal-fractional differential equations with exponential memory kernels. The proposed approach is based on a predictor–corrector framework augmented by a specially constructed iterative procedure tailored for fractal-fractional operators with exponential memory. In contrast to standard predictor–corrector methods, the iteration explicitly incorporates the structure of the exponential kernel into the correction step, which improves the numerical treatment of memory-dependent terms and enhances stability and convergence behavior. The mathematical formulation is supported by the relevant definitions of fractal-fractional derivatives and integrals with exponential kernels, together with a discussion of the stability and convergence properties of the resulting numerical scheme. To demonstrate its applicability, the method is tested on representative models arising in applied mathematics, including financial dynamics, memristor-based systems, chaotic models, and population growth processes. These examples are chosen to illustrate the ability of the scheme to handle nonlocal memory effects and nonlinear dynamics, rather than to provide an exhaustive range of applications. Numerical results show that the proposed method produces accurate and stable approximations and exhibits favorable convergence behavior when compared with existing predictor–corrector-type approaches for fractal-fractional equations. The study indicates that the scheme provides a reliable computational tool for the analysis of fractal-fractional models with exponential memory kernels in applied science and engineering.