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We study the propagation of a relativistic ultrashort laser pulse in two-dimensional inhomogeneous plasma with a density ripple based on the theoretical analysis and the particle-in-cell simulation. An analytical solution of the laser pulse propagating in the homogeneous plasma is completely obtained. It is found that the stable propagation of the relativistic ultrashort laser pulse in homogeneous plasma can be realized, and the propagation distance and intensity of the laser pulse and the spatial distribution of electron density can be effectively modulated by adjusting the pulsewidth of the incident laser and the plasma frequency. More interestingly, in the inhomogeneous plasma with a density ripple, when the wave amplitude of the density ripple is less than a critical value, the intensity of the ultrashort laser pulse is almost unchanged and the propagation characteristics of the laser in the inhomogeneous plasma are basically consistent with those in the homogeneous plasma. However, when the wave amplitude of the density ripple is larger than the critical value, the inhomogeneity of the plasma has an obvious effect on the laser characteristics, and the intensity and the spatial distribution of the laser pulse will be modulated by the plasma density ripple. The influence of the laser pulsewidth, plasma density, and plasma density ripple on the laser characteristics is discussed in detail. [ABSTRACT FROM AUTHOR]