Let $S \to X$ be an $S^3$-fiber bundle over a smooth manifold $X$. If $S$ is an oriented manifold does this fiber bundle admit the structure of an $SU(2)$-principal bundle?


There is a similar theorem for the case of circle bundles and is proved in Morita's book on differential forms. Unfortunately, I do not see a way to extend his argument to this case and he does not discuss $SU(2)$-principal bundles in detail.



Does an oriented $S^3$ fiber bundle admit the structure of a principal $SU(2)$-bundle?
No. (The main idea here is present in Dylan Wilson's comment.)


Every principal $SU(2)$-bundle over $S^2$ is trivial, because $\pi\_1 SU(2)$ is trivial. But there is a nontrivial oriented bundle over $S^2$ with fiber $S^3$, namely the unit sphere bundle of the nontrivial rank $4$ vector bundle. (There are precisely two rank $r$ oriented vector bundles over $S^2$ if $r\ge 3$, because $\pi\_1 SO(r)$ has order two.)


I should explain why the nontrivial vector bundle has nontrivial unit sphere bundle, that is, why $\pi\_1 SO(4)$ injects into $\pi\_1 SDiff(S^3)$. 


You can use the big theorem of Hatcher ("Smale Conjecture"), which says that $\pi\_k SO(4)$ maps isomorphically to $\pi\_k SDiff(S^3)$ for all $k$. 


Alternatively, you can use that the bundle is detected by the Stiefel-Whitney class $w\_2$, and that Stiefel-Whitney classes of vector bundles are invariant under fiber homotopy equivalence of unit sphere bundle.


Or you can use $\pi\_3$ instead of $\pi\_1$ as suggested by Dylan Wilson, making a bundle over $S^4$; there are elements of $\pi\_3 SO(4)\cong \mathbb Z\times \mathbb Z$ not coming from $\pi\_3 SU(2)\cong\mathbb Z$. Again, the resulting sphere bundle is nontrivial either by Hatcher's theorem or by using characteristic classes. I'm not sure what the most elementary version of the characteristic-class argument would be. 


One example of a rank $4$ real vector bundle over $S^4$ that does not admit a complex structure is the tangent bundle!


