Yes, every simply-connected rational homology sphere is topologically the $4$-sphere. Simply-connected closed topological $4$-manifolds are classified by their intersection form $Q_X:H^2(X;\Bbb Z) \times H^2(X ;\Bbb Z) \to \Bbb Z$ and their Kirby-Siebenmann invariant by a famous theorem of Freedman. If the form is even, the KS invariant automatically vanishes. If $X$ is a rational homology sphere, $Q_X$ clearly vanishes (as $H^2(X;\Bbb Z)=0$), and therefore $X$ must be homeomorphic to the $4$-sphere.

See: Michael H. Freedman & Frank Quinn Topology of 4-Manifolds (PMS-39)

Yes, every simply-connected rational homology $4$-sphere is topologically the $4$-sphere. Simply-connected closed topological $4$-manifolds are classified by their intersection form $Q_X:H^2(X;\Bbb Z) \times H^2(X ;\Bbb Z) \to \Bbb Z$ and their Kirby-Siebenmann invariant by a famous theorem of Freedman. If the form is even, the KS invariant automatically vanishes. If $X$ is a rational homology sphere, $Q_X$ clearly vanishes (as $H^2(X;\Bbb Z)=0$), and therefore $X$ must be homeomorphic to the $4$-sphere.

See: Michael H. Freedman & Frank Quinn Topology of 4-Manifolds (PMS-39)