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.
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