Curvature represents tidal effects, and the equivalence principle is by definition only applicable to regions of spacetime where tidal effects are too small to notice. So the equivalence principle on its own cannot tell you about curvature.
However, if you couple the equivalence principle with some basic observations then you can infer that curvature is needed. Specifically, the equivalence principle implies that an object in free fall has a worldline which is locally straight (a geodesic). Now, couple this with observations of tidal effects. For example, consider two objects dropped from rest at the same altitude over a planet. Initially they are at rest with respect to each other, then the distance between them reduces.
So, per the equivalence principle we have two straight lines. Since they are initially at rest their world lines are initially parallel. And yet the distance between them reduces, in principle (drill holes through the planet) they will intersect at the center of the planet.
This combination (straight initially parallel lines that intersect) is not possible in flat spacetime, but is possible if spacetime is curved.