Thin films of the ferromagnetic metal SrRuO3 (SRO) show a varying easy magnetization axis depending on the epitaxial strain, and undergo a metal-to-insulator transition with decreasing film thickness. We have investigated the magnetic properties of SRO thin films with varying thicknesses fabricated on SrTiO3(001) substrates by soft x-ray magnetic circular dichroism at the Ru M-2,M-3 edge. Results have shown that, with decreasing film thickness, the film changes from ferromagnetic to nonmagnetic at around 3 monolayer thickness, consistent with previous magnetization and magneto-optical Kerr effect measurements. The orbital magnetic moment perpendicular to the film was found to be similar to 0.1 mu(B)/Ru, and remained nearly unchanged with decreasing film thickness while the spin magnetic moment decreases. A mechanism for the formation of the orbital magnetic moment is discussed based on the electronic structure of the compressively strained SRO film.