The space is not an affine variety. A smooth affine variety of dimension $n$ is a Stein manifold, and thus must be homotopy equivalent to an $n$-dimensional CW complex (see Theorem 7.2 of Milnor's Morse Theory). However, $SL_2(\mathbb{C})/SL_2(\mathbb{Z})$ is not homotopy equivalent to a 3-complex.

Notice $SL_2(\mathbb{C})/SL_2(\mathbb{Z})$ is finitely covered by e.g. $SL_2(\mathbb{C})/\Gamma(4)$, which therefore has fundamental group $\Gamma(4)$ which is a free group. There is a fibration $S^3=SU(2) \to SL_2(\mathbb{C})/\Gamma(4) \to \mathbb{H}^3/\Gamma(4)$. Take any homologically non-trivial simple closed loop in $\mathbb{H}^3/\Gamma(4)$, then the induced $S^3$ bundle over $S^1$ is trivial since it is orientable, so we see an essential $S^3\times S^1\subset SL_2(\mathbb{C})/\Gamma(4)$ (in fact, being a bit more careful, there is a retract $SL_2(\mathbb{C})/\Gamma(4)\to S^3\times S^1$). Therefore the cohomological dimension of $SL_2(\mathbb{C})/\Gamma(4)$ is 4 (one also sees it is homotopy equivalent to a 4-complex since $\mathbb{H}^3/\Gamma(4)$ is homotopy equivalent to a wedge of loops), so it is not homotopy equivalent to a 3-complex, and thus neither is $SL_2(\mathbb{C})/SL_2(\mathbb{Z})$.