Cylindrical gears are used extensively due to their significant advantages including high efficiency, high load-bearing capacity, and long lifespan. However, the machining accuracy of cylindrical gears is significantly affected by motion errors and force-induced errors of machine tools. In this study, a motion error model of the machine tools was established based on multi-body system theory and homogeneous coordinate transformation method, quantifying the contributions and variation patterns of 12 key errors in the A and B-axes to workpiece geometric errors. Then, by using the stiffness analytical model and the spatial meshing theory, the influence of the force-induced elastic deformation of the shaft of rolling wheel and the springback of the workpiece tooth flank on the geometric error was revealed. Finally, taking the through rolling of a spur cylindrical gear with a module of 1.75 mm, a pressure angle of 20°, and 46 teeth as an example, the force-induced elastic deformation model of the shaft was verified by the rolling tests. Results show that for 40CrNiMo steel, the total profile deviation, total helix deviation, and single pitch deviation in the X-direction caused by rolling forces are 32.48 μm, 32.13 μm, and 32.13 μm, respectively, with a maximum contact rebound is δ_c = 28.27 μm. The relative error between theoretical and measured X-direction spindle deformation is 8.26%. This study provides theoretical foundation and experimental support for improving the precision of rolling process.