A proof of this can be extracted from Steinberg's paper "Générateurs, relations et revêtements de groupes algébriques". If Ive understood it correctly, he shows that a Cartan subgroup of the universal central extension of $SL_2(K)$ (for a field $K$ with at least 4 elements) is generated by elements $h(t)$ for $t$ a nonzero element of the field, that in particular satisfy $h(tu^2)=h(t)h(u^2)$, and the center of the universal central extension is the kernel of the map from the Cartan subgroup to $K^*/\pm 1$ taking $h(t)$ to $t$.  In the case when $K$ is the reals, this kernel is generated by $h(-1)$ as every element is a square or a square times $-1$. So the center of the universal central extension is  generated by $h(-1)$ and in particular is cyclic.   The center is known be be at least $Z$, so the center of the universal central extension is exactly $Z$, and the universal central extension is therefore the same as the universal cover.