Are there more embeddings $U(2) \hookrightarrow SO(4)$? It is easy to prove that $SO(4)$ acts transitively and freely on $S^2$ with fiber $U(2)$. Therefore, we can identify each point of $S^2$ with a particular embedding $U(2) \hookrightarrow SO(4)$.
My question is ¿are there more embeddings $U(2) \hookrightarrow SO(4)$ or are they completely parametrized by $S^2$?
 A: I'm taking "embedding" to mean "homomorphism of groups which happen to also be embeddings of manifolds".  If you meant "embedding" in the weaker sense of forgetting the group structure, then as Olivier says in the comments, the problem is very difficult.  Also, the action of $SO(4)$ on $S^2$ is transitively, but definitely not free, since, as you noted, the isotropy subgroup is $U(2)\neq \{e\}$. 
With that out of the way, the answer is that you've found half of the embeddings $U(2)\rightarrow SO(4)$.  More precisely, call two embeddings conjugate if their images are conjugate in $SO(4)$.  Then, the U(2)s you've described completely encompass one conjugacy class of embedding, and there is precisely one other conjugacy class of embeddings.
The two conjugacy classes of embeddings can be seen in the following way:  Given the double cover $SU(2)\times SU(2)\rightarrow SO(4)$, the images of $SU(2)\times S^1$ and $S^1\times SU(2)$ (where, say, $S^1 = \operatorname{diag}(z,\overline{z})$) are non-conjugate $U(2)$s.  To show that, up to conjugacy, these are the only $U(2)$s requires (I think) a bit of representation theory.
Now, all the $U(2)$s you've found arise as the stabilizers of the natural action of $SO(4)$ on $S^2$.  Now, suppose you've found, once and for all, the particular $U(2)$ which stabilizes a particular point $p\in S^2$ in the sense that $A\ast p = p$ for all $A\in U(2)$.  Pick any matrix $B\in SO(4)$.  A quick calculation shows that the stabilizer of the point $B\ast p$ is $B U(2) B^{-1}$.  In particular, all the $U(2)$ you've found are conjugate.
Conversely, given any conjugate $CU(2)C^{-1}$ of the $U(2)$ you found, this is the stabilizer of the point $C\ast p$.  Thus, your $U(2)$s which are paramaterized by $S^2$ have provided a complete description of all the $U(2)$s in a particular conjugacy class.
