As discussed in the comments, the real part of the integral diverges due to the singularity near the origin and we can redefine it using the principal value in order to get a finite result. So let us consider
$$ I (G) = \operatorname{P.V.} \int \limits_{-\infty}^\infty \frac{\mathrm{e}^{\mathrm{i} G r} \operatorname{erf}(r)}{r^2} \, \mathrm{d} r $$
for $G \in \mathbb{R}$ instead.
The real part is
$$ \operatorname{Re} I(G) = \lim_{\varepsilon \to 0^+} \left[~\int \limits_{-\infty}^{-\varepsilon} \frac{\cos(G r) \operatorname{erf}(r)}{r^2} \, \mathrm{d} r + \int \limits_\varepsilon^\infty \frac{\cos(G r) \operatorname{erf}(r)}{r^2} \, \mathrm{d} r \right] .$$
Since the integrand is an odd function, the first integral is exactly the negative of the second (just let $r \to -r$) and they cancel, so $\operatorname{Re} I(G) = 0$.
The imaginary part of the integrand is regular at the origin, so we can drop the principal value and write
$$ \operatorname{Im} I(G) = \int \limits_{-\infty}^\infty \frac{\sin(G r) \operatorname{erf}(r)}{r^2} \, \mathrm{d} r \, . $$
Clearly, $\operatorname{Im} I(0) = 0$. Now let $G \in \mathbb{R} \setminus \{0\}$. By definition we have
$$ \operatorname{erf} (r) = \frac{2}{\sqrt{\pi}} \int \limits_0^r \mathrm{e}^{- t^2} \mathrm{d} t = \frac{2}{\sqrt{\pi}} r \int \limits_0^1 \mathrm{e}^{- r^2 u^2} \mathrm{d} u $$
for $r \in \mathbb{R}$. Fubini's theorem allows us to interchange the two integrals to obtain
$$ \operatorname{Im} I(G) = \frac{2}{\sqrt{\pi}} \int \limits_0^1 \int \limits_{-\infty}^\infty \frac{\sin(Gr)}{r} \mathrm{e}^{-u^2 r^2} \mathrm{d} r \, \mathrm{d} u \, . $$
The inner integral can be found by differentiating it with respect to $G$, computing the resulting Fourier transform of a Gaussian and then integrating with respect to $G$ again (in fact, I have recently done this calculation in this answer to another question). The result is $\pi \operatorname{erf} \left(\frac{G}{2u}\right)$, so
\begin{align}
\operatorname{Im} I(G) &= 2 \sqrt{\pi} \int \limits_0^1 \operatorname{erf} \left(\frac{G}{2u}\right) \, \mathrm{d} u \overset{\text{IBP}}{=} 2 \sqrt{\pi} \operatorname{erf} \left(\frac{G}{2}\right) + 2 G \int \limits_0^1 \frac{\mathrm{e}^{-\frac{G^2}{4u^2}}}{u} \, \mathrm{d} u\\ & \!\!\!\!\overset{u = \frac{\lvert G \rvert}{2 \sqrt{v}}}{=} 2 \sqrt{\pi} \operatorname{erf} \left(\frac{G}{2}\right) + G \int \limits_{G^2/4}^\infty \frac{\mathrm{e}^{-v}}{v} \, \mathrm{d}v = 2 \sqrt{\pi} \operatorname{erf} \left(\frac{G}{2}\right) + G \operatorname{E}_1 \left(\frac{G^2}{4}\right) \, .
\end{align}
Here $\operatorname{E}_1$ is an exponential integral.
Therefore, the final result is
$$ I(G) = \begin{cases} \mathrm{i} \left[2 \sqrt{\pi} \operatorname{erf} \left(\frac{G}{2}\right) + G \operatorname{E}_1 \left(\frac{G^2}{4}\right) \right]& \text{for} ~ G \in \mathbb{R} \setminus \{0\} \\ 0 & \text{for} ~ G = 0 \end{cases}. $$
