Assuming that the boundary of $D$ is regular, say $D$ is a bounded domain with Lipschitz boundary, **the density is true.** Equivalent definition of $H^2$ is as follows: $H^s=W^{s,2}$, where
$$
\Vert f\Vert_{W^{2,s}}=\Vert f\Vert_2+\left(\int_{\mathbb{R}^d}\int_{\mathbb{R}^d}
\frac{|f(x)-f(y)|^2}{|x-y|^{n+2s}}\right)^{1/2}
$$
For a proof, see Section 3 in [1]. Now it is relatively easy to prove that a function in $W^{2,s}$ that vanishes in the complement of $D$ can be approximated by $C_0^\infty(D)$. This is more or less contained in Corollary 5.5 in [1].

I am pretty sure that for general domains there are counterexample. If you are interested in irregular domains I can try to find one.

**[1] Di Nezza, E., Palatucci, G., Valdinoci, E.,** Hitchhiker's guide to the fractional Sobolev spaces.. *Bull. Sci. Math.* 136 (2012), no. 5, 521-573.
(MathSciNet review).