Rotavirus remains a leading cause of gastroenteritis in children under five in low- and middle-income countries due to waning immunity and incomplete vaccine coverage. To address this, we propose a mathematical model to analyze the transmission dynamics with primary and booster vaccination strategies. The model is formulated using the fractal-fractional derivative in the Caputo-Fabrizio sense, which allows for the incorporation of memory effects and hereditary properties in disease evolution. The population is structured into five compartments, including booster-immunized individuals. We derive the disease-free and endemic equilibrium points and analyze their local stability. The basic reproduction number is computed to determine the threshold conditions for disease persistence. We establish the existence and Hyers-Ulam (H-U) stability of the model, and validate the results through numerical simulations using the Adams-Bashforth method (ABM), confirmed by comparison with Runge-Kutta 4th Order (RK-4) solution plots to assess the booster vaccination impact. The results reveal that booster immunization plays a significant role in reducing the infection burden, thereby highlighting its relevance in public health planning.