In the context of image smoothing (e.g. scale space), the Gaussian is the only rotationally symmetric separable* kernel.

That is, if we require $$F[x,y]=f[x]f[y]$$ where $[x,y]=r[\cos\theta,\sin\theta]$, then rotational symmetry requires \begin{align} F_\theta &= f'[x]f[y]x_\theta+f[x]f'[y]y_\theta \\ &= -f'[x]f[y]y+f[x]f'[y]x = 0 \\ &\implies \\ \frac{f'[x]}{xf[x]} &= \frac{f'[y]}{yf[y]} = \mathrm{const.} \end{align} which is equivalent to $\log\big[f[x]\big]'=cx$.

Requiring that $f[x]$ be a proper kernel then requires the constant be negative and the initial value positive, yielding the Gaussian kernel.

*In the context of probability distributions, separable means independent, while in the context of image filtering it allows the 2D convolution to be reduced computationally to two 1D convolutions.