Many methods based on acoustic vibration characteristics have been studied to indirectly assess fruit ripeness via fruit firmness. Among these, the frequency of the 0S2 vibration mode measured on the equator has been examined, but soft-flesh fruit do not show the 0S2 vibration mode. In this study, a Rayleigh wave is generated on a soft mango fruit using the impulse excitation force generated by a laser-induced plasma shock wave technique. Then, the flesh firmness of mangoes is assessed in a non-contact and non-destructive manner by observing the Rayleigh wave propagation velocity because it is correlated with the firmness (shear elasticity), density, and Poisson's ratio of an object. If the changes in the density and Poisson's ratio are small enough to be ignored during storage, then the Rayleigh wave propagation velocity is strongly correlated to fruit firmness. Here, we measure the Rayleigh wave propagation velocity and investigate the effect of storage time. Specifically, we investigate the changes in firmness caused by ripening. The Rayleigh wave propagation velocity on the equator of Kent mangoes tended to decrease by over 4% in 96 h. The Rayleigh wave measured on two different lines propagated independent distance and showed a different change rate of propagation velocity during 96-h storage. Furthermore, we consider the reliability of our method by investigating the interaction of a mango seed on the Rayleigh wave propagation velocity.