  Using data from the Fermi Gamma-Ray Space Telescope, a spatially extended
component of gamma rays has been identified from the direction of the Galactic
Center, peaking at energies of ~2-3 GeV. More recently, it has been shown that
this signal is not confined to the innermost hundreds of parsecs of the Galaxy,
but instead extends to at least ~3 kpc from the Galactic Center. While the
spectrum, intensity, and angular distribution of this signal is in good
agreement with predictions from annihilating dark matter, it has also been
suggested that a population of unresolved millisecond pulsars could be
responsible for this excess GeV emission from the Inner Galaxy. In this paper,
we consider this later possibility in detail. Comparing the observed spectral
shape of the Inner Galaxy's GeV excess to the spectrum measured from 37
millisecond pulsars by Fermi, we find that these sources exhibit a spectral
shape that is much too soft at sub-GeV energies to accommodate this signal. We
also construct population models to describe the spatial distribution and
luminosity function of the Milky Way's millisecond pulsars. After taking into
account constraints from the observed distribution of Fermi sources (including
both sources known to be millisecond pulsars, and unidentified sources which
could be pulsars), we find that millisecond pulsars can account for no more
than ~10% of the Inner Galaxy's GeV excess. Each of these arguments strongly
disfavor millisecond pulsars as the source of this signal.
