http://en.wikipedia.org/wiki/Gravitation
No, the acceleration of the object, regardless of its mass, is entirely due to the spacetime curvature in which it is falling...neglecting air resistance and such, all objects, whether they weight 1 gram or a million will fall in Earth's gravitational field/spacetime curvature (for example), at 9.81m/s^2. As they will also fall at a constant rate (given that the field curvature doesn't go off the scale, like in a black hole), in whatever gravitational field they find themselves in. So, a 1 gram object, will fall at the same rate as a 1 million gram object. Both will hit the ground at the same time. GR is not contradicted at all. All GR is saying is that you cannot tell the difference between a gravitational field produced via acceleration or naturally occurring spacetime curvature. Since sitting on the Earth's surface, you are experiencing an
acceleration towards the body of the Earth of 1g (9.81m/s^2), GR is entirely correct in its assumptions. Hop in your car and slam the foot down on the accelerator...accelerate at roughly 19.5m/s^2 and you'll feel the force of 2g's pushing you into the back of your seat. Then go to a planet with that gravitational pull....you'll feel exactly the same, twice as heavy as you normally would feel.
Frame dragging will cause a minute change in the trajectory of the falling object, for most gravitational fields. Only in extreme cases around objects like neutron stars and black holes will you get enough frame dragging to cause an appreciable change in the trajectory of the falling objects.