A Riemannian manifold is called geometrically formal if the wedge product of any two harmonic forms is again harmonic. We classify geometrically formal compact 4-manifolds with nonnegative sectional curvature. If the sectional curvature is strictly positive, the manifold must be homeomorphic to $S^4$ or to $CP^2$. In particular, the Hopf conjecture on $S^2 \times S^2$ holds in the class of geometrically formal manifolds. If the sectional curvature is strictly positive and we relax the condition of geometric formality to the requirement that the length of harmonic 2-forms is not too nonconstant, then the manifold must be homeomorphic to $S^4$ or to a connected sum of $CP^2$s.