A reliable seismic vulnerability assessment of historical masonry constructions is essential for guiding conservation strategies for architectural heritage. Such assessments rely on accurately predicting the structural response under seismic loading. Static pushover analysis is a widely recognized tool in current building codes for this purpose, as it captures nonlinear structural behavior while allowing seismic input to be represented via a design spectrum. This work proposes an extension of the Non-Smooth Contact Dynamics (NSCD) framework to the static pushover analysis of historical masonry constructions. The NSCD method is a time-stepping approach developed to investigate the dynamic behavior of discrete systems of rigid or deformable bodies. By considering a discrete time-integrated form of the equations of motion, describing hard-contact laws in a time-averaged sense, and introducing suitable contact gaps, the NSCD method is able to advance the solution in time with a fixed time step, irrespective of non-smooth events such as collisions between bodies. Several applications of the NSCD method have proven its effectiveness for historical masonry structures modeled as systems of rigid blocks in unilateral-frictional contact. More recently, a variational NSCD approach has been proposed, exploiting a convex second-order conic programming (SOCP) formulation of the time-step problem to address large-scale 3D masonry block structures under seismic excitation. Here, it is shown that such a variational NSCD framework can be extended to perform static pushover analyses by treating time as a monotonically increasing parameter and neglecting inertia, so that the analysis is formulated as a sequence of quasi-static SOCP equilibrium problems under increasing lateral loads. Using a suitable collision-detection algorithm, this approach accurately models the unilateral-frictional contact behavior and enables the computation of structural capacity curves and collapse mechanisms. Numerical results demonstrate the computational potential of the method and its merit as a robust, practical tool for the seismic assessment of historical masonry structures.