  Radio images of the Galactic Center supermassive black hole, Sagittarius A*
(Sgr A*), are dominated by interstellar scattering. Previous studies of Sgr A*
have adopted an anisotropic Gaussian model for both the intrinsic source and
the scattering, and they have extrapolated the scattering using a purely
$\lambda^2$ scaling to estimate intrinsic properties. However, physically
motivated source and scattering models break all three of these assumptions.
They also predict that refractive scattering effects will be significant, which
have been ignored in standard model fitting procedures. We analyze radio
observations of Sgr A* using a physically motivated scattering model, and we
develop a prescription to incorporate refractive scattering uncertainties when
model fitting. We show that an anisotropic Gaussian scattering kernel is an
excellent approximation for Sgr A* at wavelengths longer than 1cm, with an
angular size of $(1.380 \pm 0.013) \lambda_{\rm cm}^2\,{\rm mas}$ along the
major axis, $(0.703 \pm 0.013) \lambda_{\rm cm}^2\,{\rm mas}$ along the minor
axis, and a position angle of $81.9^\circ \pm 0.2^\circ$. We estimate that the
turbulent dissipation scale is at least $600\,{\rm km}$, with tentative support
for $r_{\rm in} = 800 \pm 200\,{\rm km}$, suggesting that the ion Larmor radius
defines the dissipation scale. We find that the power-law index for density
fluctuations in the scattering material is $\beta < 3.47$, shallower than
expected for a Kolmogorov spectrum ($\beta=11/3$), and we estimate $\beta =
3.38^{+0.08}_{-0.04}$ in the case of $r_{\rm in} = 800\,{\rm km}$. We find that
the intrinsic structure of Sgr A* is nearly isotropic over wavelengths from
1.3mm to 1.3cm, with a size that is roughly proportional to wavelength. We
discuss implications for models of Sgr A*, for theories of interstellar
turbulence, and for imaging Sgr A* with the Event Horizon Telescope.
