A possible experiment is to place several small objects randomically at different points, floating inside the ship.
Without a gravitational field, any small initial speed from the instant of releasing any of them, continues until it kicks a wall. It bounces and follows another direction. The randominess of the positions of the objects at any instant tends to be the same.
Inside a gravitational field, and because that field is always non uniform, each object follows its own geodesic, that is different from the geodesic of the centre of gravitation of the ship.
For example, their average speed is the same of the ship, due to the randominess hypothesis. That ones floating in the half closer to the star or planet tends to become even closer, because the average speed is below what is required for orbiting. The opposite happens to the other half.
They are, we can say, expelled from the middle, and concentrates closer to the opposite walls, one closer and the other more distant from the gravity source.
Of course it takes time to that effect becomes noticeable, so it doesn't contradicts the equivalence principle.