  The Galactic Center (GC) region hosts a variety of powerful astronomical
sources and rare astrophysical processes that emit a large flux of non-thermal
radiation. The inner 375 pc x 600 pc region, called the Central Molecular Zone,
is home to the supermassive black hole Sagittarius A*, massive cloud complexes,
and particle accelerators such as supernova remnants. We present the results of
our improved analysis of the very-high-energy (VHE) gamma-ray emission above 2
TeV from the GC using 125 hours of data taken with the VERITAS
imaging-atmospheric Cherenkov telescope between 2010 and 2018. The central
source VER J1745-290, consistent with the position of Sagittarius A*, is
detected at a significance of 38 standard deviations above the background level
$(38\sigma)$, and we report its spectrum and light curve. Its differential
spectrum is consistent with a power law with exponential cutoff, with a
spectral index of $2.12^{+0.22}_{-0.17}$, a flux normalization at 5.3 TeV of
$1.27^{+0.22}_{-0.23}\times 10^{-13}$ TeV-1 cm-2 s-1, and cutoff energy of
$10.0^{+4.0}_{-2.0}$ TeV. We also present results on the diffuse emission near
the GC, obtained by combining data from multiple regions along the GC ridge
which yield a cumulative significance of $9.5\sigma$. The diffuse GC ridge
spectrum is best fit by a power law with a hard index of 2.19 $\pm$ 0.20,
showing no evidence of a cutoff up to 40 TeV. This strengthens the evidence for
a potential accelerator of PeV cosmic rays being present in the GC. We also
provide spectra of the other sources in our field of view with significant
detections, composite supernova remnant G0.9+0.1 and HESS J1746-285.
