This study presents a novel mathematical model of a generalized magnetothermoelastic half-space based on the Green–Naghdi theorem, namely type-I and type-III. The half-space surface undergoes ramp-type heating and is positioned on a sturdy base to prevent movement. This research is novel as it employs Caputo’s definition of fractional derivatives within the context of Maxwell’s time-fractional equations. Laplace transform methods are used to obtain the solutions. Tzou’s iterative method has been used to calculate inversions of the Laplace transform. The findings include quantitative answers for temperature increase, strain, displacement, stress, induced magnetic field, and induced electric field distributions. The time-fraction parameter defined by Maxwell’s equation considerably influences all essential mechanical functions, but the thermal functions remain unchanged. In Maxwell’s equations, the time-fractional parameter functions augment the induced electric field inside the material, acting as a resistor to particle motion and the induced magnetic field, while concurrently facilitating the induced electric field. Moreover, the thermal, mechanical, and magnetoelectric waves of Green–Naghdi type-III propagate at a reduced velocity compared to type-I. The fundamental magnetic field substantially influences all examined functions.