  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.
