We perform collisionless N-body simulations to investigate whether binary mergers between rotationally supported dwarfs can lead to the formation of dwarf spheroidal galaxies (dSphs). Our simulation campaign is based on a hybrid approach combining cosmological simulations and controlled numerical experiments. We select merger events from a Constrained Local Universe simulation of the Local Group (LG) and record the properties of the interacting dwarf-sized halos. This information is subsequently used to seed controlled experiments of binary encounters between dwarf galaxies consisting of exponential stellar disks embedded in cosmologically motivated dark matter halos. These simulations are designed to reproduce eight cosmological merger events, with initial masses of the interacting systems in the range ~(5-60) × 107 M sun, occurring quite early in the history of the LG, more than 10 Gyr ago. We compute the properties of the merger remnants as a distant observer would and demonstrate that at least three of the simulated encounters produce systems with kinematic and structural properties akin to those of the classic dSphs in the LG. Tracing the history of the remnants in the cosmological simulation to z = 0, we find that two dSph-like objects remain isolated at distances >~ 800 kpc from either the Milky Way or M31. These systems constitute plausible counterparts of the remote dSphs Cetus and Tucana which reside in the LG outskirts, far from the tidal influence of the primary galaxies. We conclude that merging of rotationally supported dwarfs represents a viable mechanism for the formation of dSphs in the LG and similar environments.