Large earthquakes are commonly followed by abundant aftershocks that are densely located around the coseismic rupture zone. Laboratory experiments indicate that "microscopic" brittle rock failures (acoustic emission) are associated collectively with a "macroscopic" damage-related inelastic relaxation. Utilizing basic relations between local brittle failures and gradual inelastic strain in a viscoelastic damage rheology model, we develop connections between aftershock decay rates and the aftershocks-induced component of geodetic deformation. The discussed mechanism is relevant for postseismic relaxation produced by sources located within the seismogenic zone, and especially in regions that overlap locations of high aftershocks activity. Assuming the Omori-Utsu decay rate for aftershocks, we find that the temporal decay of the damage-related postseismic relaxation follows a generalized power-law relation with the standard Omori-Utsu law as a limit case. The results provide a way for estimating the separate contributions to observed postseismic displacements that stem from brittle failures in the seismogenic zone (aftershocks) and other (aseismic) processes. Using the obtained theoretical expectations, we analyze postseismic displacements measured by GPS stations around the North Anatolian fault similar to 3 months following the 1999 M7.4 Izmit earthquake. We find that the observed postseismic displacements decay slower than the aftershock seismicity. Based on our theoretical results, we conclude that up to 50% of the measured surface displacements at near-fault sites can be attributed to aftershock-induced inelastic deformation in the seismogenic zone. The remainder postseismic deformation can generally be explained by relaxation in the deeper ductile substrate.