Using first-principles calculations, we predict a strong coupling of lattice, charge, and spin degrees of freedom in Ruddlesden-Popper phase ${\mathrm{La}}_{3}{\mathrm{Ni}}_{2}{\mathrm{O}}_{7}$, which enables a phase control with epitaxial strain. While the bulk ground state is metallic, moderate compressive strain is found to trigger a Peierls transition to an insulating state in concurrence with a breathing distortion of ${\mathrm{NiO}}_{6}$ octahedra. The Peierls transition is microscopically interpreted as a band-gap opening arising from lifting and lowering of the Ni $d\ensuremath{-}{e}_{g}$ states due to the octahedral breathing.