Friction stir spot welding (FSSW) of similar AA5052-H32 joints has numerous benefits in shipbuilding, aerospace, and automotive structural applications. In addition, studying the role of tool rotation speed on the microstructure features, achieved textures, and joint performance of the friction stir spot-welded (FSSWed) joint still needs more systematic research. Different FSSWed AA5052-H32 lap joints of 4 mm thickness were produced at different heat inputs using three tool rotation speeds of 1500, 1000, and 500 rpm at a constant dwell time of 2 s. The applied thermal heat inputs for achieving the FSSW processes were calculated. The produced joints were characterized by their appearance, macrostructures, microstructures, and mechanical properties (hardness contour maps and maximum tensile–shear load) at room temperature. The grain structure and texture developed for all the FSSWed joints were deeply investigated using an advanced electron backscattering diffraction (EBSD) technique and compared with the base material (BM). The main results showed that the average hardness value of the stir zone (SZ) in the welded joints is higher than that in the AA5052-H32 BM for all applied rotation speeds, and it decreases as the rotation speed increases from 500 to 1000 rpm. This SZ enhancement in hardness compared to the BM cold-rolled grain structure is caused by the high grain refining due to the dynamic recrystallization associated with the FSSW. The average grain size values of the stir zones are 11, 9, and 4 µm for the FSSWed joints processed at 1500, 1000, and 500 rpm, respectively, while the BM average grain size is 40 µm. The simple shear texture with B/-B components mainly dominates the texture. Compared to the welded joints, the joint processed at 500 rpm and a 2 s duration time attains the highest tensile-shear load value of 4330 N. This value decreases with increasing rotation speed to reach 2569 N at a rotation speed of 1500. After tensile testing of the FSSWed joints, the fracture surface was also examined and discussed.