Any mixture of Gaussians has a density, which limits then sense in which a statement like you want to make can be true. The statement you propose doesn't make sense (in part) since a distribution is not a set.

One statement which I believe is true is

Mixtures of Gaussian distributions are dense in the set of probability distributions, with respect to the weak topology.

(By "weak topology" I mean the probabilists' weak topology, also called the topology of convergence in distribution, the vague topology, and the weak-* topology.)

I haven't checked details, but this should follow since in this topology, finite linear combinations of point masses are dense, and point masses can be approximated by Gaussian distributions.

The corresponding statement about density with respect to total variation is false, since discrete distributions cannot be approximated by distributions with density in total variation.