Time integration schemes play a crucial role in the accuracy and stability of peridynamic simulations. In this work, we reinterpret the velocity Verlet algorithm, which is the most commonly used time-integration method in peridynamics, as a variational integrator (VI) and explore its implications for bond-based formulation of peridynamics. By deriving the Verlet scheme from a discrete variational principle, we highlight its structure-preserving properties, including symplecticity, momentum conservation, and long-term stability in energy evolution. These properties make it particularly suitable for peridynamic simulations, where nonlocal interactions and damage evolution present significant numerical challenges. Additionally, we address the velocity approximation used in the Verlet method. We demonstrate that the interpretation of the Verlet scheme as a variational integrator leads to a velocity approximation that differs from the typically used finite difference definition. These two approximations are compared in terms of their accuracy and energy behavior for the bond-based perdiynamics simulations. Finally, we provide an outlook on other variational integrators, both explicit and implicit, in the context of peridynamic simulations.