Elastic properties and their pressure dependence of rare gas solids (RGSs) Ne, Ar, Kr, and Xe are calculated using the ab initio full-potential linear muffin-tin-orbital method to study their systematics. Though the local density approximation (LDA) underestimates volume of these substances, especially Ne, equations of state and elastic constants are well reproduced in the entire pressure range up to a megabar. The Cauchy deviation, the Zener anisotropy ratio, and the normalized elastic constant are investigated as a function of pressure. It is found that many-body contribution to the interatomic force surprisingly increases with increasing pressure even in all RGSs and it is larger in the heavier element. The elastic anisotropy of all four RGSs gradually decreases with increasing pressure. The normalized elastic constants c(11)' and c(12)' largely deviate from the ideal values, and c(ij)''s are remarkably insensitive to pressure over 10 GPa. Moreover, no elastic instability appears under high pressure in all RGSs. Consequently, LDA is sufficiently valid even for RGS since the van der Waals interaction becomes minor with increasing pressure. These trends of high-pressure elastic property are quite different from those of ionic solid and metal.