Here, we report on the low-energy excitations within the paramagnetic spin-orbit insulator ${\mathrm{Sr}}_{3}{\mathrm{Ir}}_{2}{\mathrm{O}}_{7}{\mathrm{F}}_{2}$ studied via resonant inelastic x-ray scattering, ab initio quantum chemical calculations, and model Hamiltonian simulations. This material is a unique ${d}^{4} {\mathrm{Ir}}^{5+}$ analog of ${\mathrm{Sr}}_{3}{\mathrm{Ir}}_{2}{\mathrm{O}}_{7}$ that forms when F ions are intercalated within the SrO layers spacing the square lattice ${\mathrm{IrO}}_{6}$ bilayers of ${\mathrm{Sr}}_{3}{\mathrm{Ir}}_{2}{\mathrm{O}}_{7}$. Due to the large distortions about the ${\mathrm{Ir}}^{5+}$ ions, our computations demonstrate that a large single-ion anisotropy yields an $S=1$ ($L\ensuremath{\approx}1, J\ensuremath{\approx}0$) ground-state wave function. Weakly coupled, excitonic modes out of the ${S}_{z}=0$ ground state are observed and are well described by a phenomenological spin-orbit exciton model previously developed for $3d$ and $4d$ transition metal ions. The implications of our results regarding the interpretation of previous studies of hole-doped iridates close to ${d}^{4}$ fillings are discussed.