Dark matter (DM) in the Milky Way halo may annihilate or decay to photons, producing monochromatic gamma rays. We search for DM-induced spectral lines using 14 years of data from the Large Area Telescope onboard the Fermi Gamma-ray Space Telescope ($\textit{Fermi}$-LAT) between $10\,\mathrm{GeV}$ and $2\,\mathrm{TeV}$ in the inner Milky Way leveraging both the spatial and spectral morphology of an expected signal. We present new constraints as strong as $\langle \sigma v \rangle \lesssim 6\times 10^{-30}\, \mathrm{cm}^3/\mathrm{s}$ for the two-to-two annihilations and $\tau \gtrsim 10^{30}\,\mathrm{s}$ for one-to-two decays, representing leading sensitivity between $10\,\mathrm{GeV}$ and $\sim$$500\,\mathrm{GeV}$. We consider the implications of our line-constraints on the Galactic Center Excess (GCE), which is a previously-observed excess of continuum $\sim$GeV gamma-rays that may be explained by DM annihilation. The Higgs portal and neutralino-like DM scenarios, which have been extensively discussed as possible origins of the GCE, are constrained by our work because of the lack of observed one-loop decays to two photons. More generally, we interpret our null results in a variety of annihilating and decaying DM models, such as neutralinos, gravitinos, and glueballs, showing that in many cases the line search is more powerful than the continuum, despite the continuum annihilation being at tree level.