For which closed smooth manifolds does the action of the diffeomorphism group on the set of almost complex structures have exactly one orbit?


For example it is true for $S^2$.



Unique almost complex structure up to diffeomorphism
You were lucky to find the only possible example. If you take any manifold of dimension $\ge 4$ you can pick an almost complex structure that is integrable in some closed ball and make it non-integrable outside of it. Since complex $n$-balls have more than one holomorphic structure, we are done. And all surfaces apart from $S^2$ have moduli of complex structures.


