Taking into account the gradient change of the near-surface viscous property, we develop a finite element model of laser-generated surface acoustic wave in composite plate. The propagation characteristics of the surface acoustic wave in the composite plate are studied in detail, and the influences of the near-surface viscous modulus, thickness, and Lam constant on the attenuation characteristics of the surface acoustic wave are discussed. In addition, the propagation characteristics of the surface acoustic wave are verified by the theoretical calculations of the dispersion and attenuation curves. The results show that the near-surface viscous modulus and thickness are related to the attenuations of the surface shear wave and the Rayleigh wave, but have no influence on the propagation velocity. Furthermore, the imaginary part of the Lam constant has great influence on the attenuations of the surface shear wave and the Rayleigh wave, whereas the imaginary part of has no effect on the attenuation characteristics of the two waves, which indicates that the attenuation of the surface acoustic wave is related to the near-surface shear viscous modulus. The study gives theoretical basis for evaluating the near-surface mechanical properties of the composite plates by the laser ultrasound technique. Besides, it provides a feasible way to study the surface micro-cracks on the composite plates.