In the present paper, the stability of a nonlinear elastic cylindrical tube made of micropolar material is analyzed. It is assumed that the elastic properties of the tube vary through the wall thickness. The problem is studied for the case of axial compression of the tube under internal and external hydrostatic pressure. Applying linearization the neutral equilibrium equations have been derived, which describe the perturbed state of the tube. By solving these equations numerically the critical curves and corresponding buckling modes have been found, and the stability regions have been constructed in the planes of loading parameters (relative axial compression, relative internal or external pressure). Using these results, the influence of elastic properties, as well as the size of the tube, on the loss of stability is studied. Special attention has been paid to the analysis of how the pattern of change in elastic parameters affects the stability of a cylindrical tube made of micropolar material.