Friction stir spot welding (FSSW) technique was used to join aluminium (AA1060) to copper (C11000). The effect of process parameters on the residual stress and the full width at half maximum was studied. A copper ring was observed on both sides of the keyhole in all the fabricated FSS Welds. The surface morphology of the copper rings was examined using a scanning electron microscope; while the residual stresses were measured using the non-destructive X-ray diffraction method. The copper ring dimension in all the fabricated FSS Welds increased with the variation of the tool shoulder plunge depths; nevertheless, the welds fabricated at 1200 rpm rotation speed displayed a decrease in the dimension of the formed copper ring. In the analysed samples only compressive residual stresses were observed and highest residual stress of −116.8 MPa was obtained on the formed copper ring of the spot welds fabricated at a rotation speed of 800 rpm and 0.5 mm tool shoulder plunge depth. The presence of higher residual stress on the copper rings was due to the extrusion of copper into the aluminium sheet. Additionally, the intensity of all the peaks for the fabricated spot welds decreased in comparison to the peaks generated by the aluminium and copper base materials. Furthermore, the tool shoulder plunge depth affected the full width at half the maximum (FWHM).