This paper investigates the effect of intertube van der Waals interaction on the stability of pristine and covalently functionalized carbon nanotubes under axial compression, using molecular mechanics simulations. After regulating the number of inner layers of the armchair four-walled (5, 5)@(10, 10)@(15, 15)@(20, 20) and zigzag four-walled (6, 0)@(15, 0)@(24, 0)@(33, 0) carbon nanotubes, the critical buckling strains of the corresponding tubes are calculated. The results show that each of the three inner layers in the functionalized armchair nanotube noticeably contributes to the stability of the outermost tube, and together increase the critical strain amplitude by 155%. However, the three inner layers in the corresponding pristine nanotube, taken together, increase the critical strain of the outermost tube by only 23%. In addition, for both the pristine and functionalized zigzag nanotubes, only the (24, 0) layer, among the three inner layers, contributes to the critical strain of the corresponding outermost tube, by 11% and 29%, respectively. The underlying mechanism of the enhanced stability related to nanotube chirality and functionalization is analyzed in detail.