  Astrophysical black holes are expected to be described by the Kerr metric.
This is the only stationary, vacuum, axisymmetric metric, without
electromagnetic charge, that satisfies Einstein's equations and does not have
pathologies outside of the event horizon. We present new constraints on
potential deviations from the Kerr prediction based on 2017 EHT observations of
Sagittarius A* (Sgr A*). We calibrate the relationship between the
geometrically defined black hole shadow and the observed size of the ring-like
images using a library that includes both Kerr and non-Kerr simulations. We use
the exquisite prior constraints on the mass-to-distance ratio for Sgr A* to
show that the observed image size is within $\sim$ 10$\%$ of the Kerr
predictions. We use these bounds to constrain metrics that are parametrically
different from Kerr as well as the charges of several known spacetimes. To
consider alternatives to the presence of an event horizon we explore the
possibility that Sgr A* is a compact object with a surface that either absorbs
and thermally re-emits incident radiation or partially reflects it. Using the
observed image size and the broadband spectrum of Sgr A*, we conclude that a
thermal surface can be ruled out and a fully reflective one is unlikely. We
compare our results to the broader landscape of gravitational tests. Together
with the bounds found for stellar mass black holes and the M87 black hole, our
observations provide further support that the external spacetimes of all black
holes are described by the Kerr metric, independent of their mass.
