A process of single diffractive dissociation off nuclei is considered on a basis of solutions to the nonlinear evolution equation. The relevant saturation scales QSDA (x,x0) are determined and their dependences on Bjorken x, atomic number A, and minimal rapidity gap Y0 ≡ 1n 1/x0 are investigated. The solutions are shown to possess a geometrical scaling in a broad kinematic region for x well below x0. The ratio σdiff/σtot is computed for several nuclei. We predict that at x ≃ 10-4 this ratio is of the order 25% which is much larger compared to the one of proton. This result indicates a possibility to observe a very strong nuclear shadowing.