In general this is not true.

Let us call a manifold which is homotopy equivalent to a complex projective space a homotopy complex projective space (HCP). In dimension 6 there are $\mathbb Z$ many manifolds (up to diffeomorphism) with homotopy type of $\mathbb{CP}^3$. They are distinguished by their first Pontryagin class. In dimension $6$ we have that (under certain conditions, which are fullfilled for HCPs) if a topological manifold admits a smooth structure, then this structure is unique. Hence if two HCPs would be homeomorphic, they would be diffeomorphic, hence they would have the same first Pontryagin class. But as I mentioned above there are $\mathbb Z$ many HCPs with pairwise different first Pontryagin classes.

Edit: I misread the question. The statement below explains only, that if a homotopy complex projective space other than $\mathbb{CP}^3$ supports a complex projective structure, then the answer would be no. As far as I know, it is not known if such spaces support even a symplectic structure.

Let us call a manifold which is homotopy equivalent to a complex projective space a homotopy complex projective space (HCP). In dimension 6 there are $\mathbb Z$ many manifolds (up to diffeomorphism) with homotopy type of $\mathbb{CP}^3$. They are distinguished by their first Pontryagin class. In dimension $6$ we have that (under certain conditions, which are fullfilled for HCPs) if a topological manifold admits a smooth structure, then this structure is unique. Hence if two HCPs would be homeomorphic, they would be diffeomorphic, hence they would have the same first Pontryagin class. But as I mentioned above there are $\mathbb Z$ many HCPs with pairwise different first Pontryagin classes.