A characteristic modulation of the magnetic domain structure on the (001) plane of an Fe-12.8 at. % Ga alloy single crystal under tensile stress was observed by magneto-optic Kerr effect microscopy. The magnetic domain structure under zero applied stress was composed of magnetic domains with magnetizations in four types of <100> magnetic easy directions on the (001) plane, which were separated by straight 90° domain walls and stair-like 180° domain walls. When a tensile stress of 11.4 MPa was applied parallel to the  direction, these 90° domain walls disappeared to release the tensile stress. A stripe domain structure that was composed of straight 180° domain walls parallel to the tensile stress direction was thus formed by the inverse magnetostrictive effect. Young's modulus of the alloy below the applied 11.4 MPa tensile stress was estimated to be 32 GPa, which is 57% smaller than the corresponding value above that tensile stress value. Consequently, a significant reduction in Young's modulus is caused by the characteristic modulation of the magnetic domain structure that occurs as a result of the inverse magnetostrictive effect.