= Solution
Translation and rotation invariance allow a radial <Green function> $G(R)$, with $R=|\mathbf r-\mathbf r'|$. Outside the source, $G''+2G'/R=0$, so $G=A/R+B$. Decay at infinity gives $B=0$.
Integrate the defining equation over a small ball centred on the source. The <divergence theorem> fixes its flux:
$$
\int_{R<\varepsilon}\nabla^2G\,dV=4\pi\varepsilon^2G'(\varepsilon)=-4\pi A=-1.
$$
Consequently
$$
\boxed{G(\mathbf r,\mathbf r')=\frac1{4\pi|\mathbf r-\mathbf r'|}.}
$$
This is the <Green function of the Laplacian> with the stated negative-delta sign. A homogeneous $1/R$ addition would change the delta normalization, so the singular coefficient is fixed, not an arbitrary integration constant.
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