This question, Is there a weak homotopy equivalence between $\mathrm{Sp}(2n,\mathbb{C})/\mathrm{U}(n)$ and $\mathrm{SU}(n)$?, is at the end of a long string of my comments in
Is Sp(2N,$\mathbb{C}$)/U(N) isomorphic to SU(N)?
And perhaps the discussion there is useful for this question. I ask this because I don't know how to show this weak homotopy equivalence. I will continue to try to answer this question on my own while I wait for responses but also thought I would just ask with different, more relevant tags.
No. $Sp(2N,\mathbb{C})$ 2-connected (you can prove this by induction on $N$; see for instance my answer at $\mathrm{Sp}(4, \mathbb{C})$ is simply connected which also works to show it is 2-connected). So by the long exact sequence in homotopy groups, $\pi_2(Sp(2N,\mathbb{N})/U(N))\cong \pi_1(U(N))\cong\mathbb{Z}$. Since $\pi_2(SU(N))$ is trivial (which can be proved by induction on $N$ in a similar way) this means they are not weak homotopy equivalent.