In view of lemma 9 of your answer it suffices to prove that the orbit equivalence relation $\Gamma$ is closed:
Let $G$ be a compact group acting continuously by homeomorphisms on a Hausdorff space $X$. Then the orbit equivalence relation $\Gamma \subset X \times X$ is closed.
Suppose $(x_i, y_i) \to (x,y)$ is a convergent net in $X \times X$ with $(x_i,y_i) \in \Gamma$. Then $x_i = g_i y_i$ for some net $g_i \in G$. Since $G$ is compact, there is a subnet $g_j$ which converges, say $g_j \to g$. Since $y_j \to y$ and $g_j \to g$ we have $x_j = g_j y_j \to gy$. But by assumption $x_j \to x$, so $x = gy$ because $X$ is Hausdorff and hence $(x,y) \in \Gamma$.