If true, such a statement would immediately imply the Poincare conjecture.  Indeed, suppose X were a closed, simply connected 3-manifold with a Mobius structure.  Near any point of $X$, there would then be a Mobius map from $X$ to $S^3$.  Analytically continuing this map (see [Thurston's notes][1], chapter 3) would give a local homeomorphism $\phi: X \to S^3$.  Since both spaces are compact, $\phi$ would be a covering map.  Since $S^3$ is simply connected it would be a homeomorphism.  A similar argument applies to projective structures, with $S^3$ replaced by $\mathbb{RP}^3$.

  [1]: http://www.msri.org/publications/books/gt3m/