Addendum: Perhaps I should expand the my answer a bit, since it wasn't terribly enlightening. In very rough terms on the algebraic side one proceeds as follows. One first needs to prove that the set of stable vector bundles with fixed topological type $V$ form a bounded family. This already uses the stability condition in an essential way.From this one deduces the existence of a scheme $Q$, usually a subscheme of a Quot scheme, such that $X\times Q$ carries a vector $E$ such that all the stable bundles in question occur among the fibres $E_q$. The moduli space $M$ would then be quotient $M=Q/G$ for some appropriate reductive $G$ acting by "change of basis". At this point, to apply GIT,one needs to know that stability in the abstract is related to to stability of the $G$-action. Note that $E$ typically won't descend i.e. $M$ need not be fine.
On the analytic side, one can form the quotient $N$ of the set of stablecomplex structures $S$ on $V$ modulo gauge equivalence. Again stability comes into playto guarantee that $N$ is reasonable. So then one gets a map ofspaces $Q^{an}\to S$ induced by $E$. This should descend to a map $M^{an}\to N$. To check that this is an analytic equivalence, one would need a description of the local analytic structures on both sides. Fortunately one has this this, see theorem 4.5.1 of [Huybrechts-Lehn]and prop 6.4.3 of [DK]. Making this into a proof would take a lot of work of course.
