Eric, Yes, plants do respond gravity. Living things that are sensitive to gravity are said to be geotropic.
Plant stems are negatively geotropic and the roots are positively geotropic.
Some scientists think this tropism is caused by plant growth hormones known as auxins. Auxins can speed up or slow down the growth of certain areas of a plant. If a plant stem falls and continues growing it is thought that the stem bends upward when auxins are pulled to the bottom side of the stem by gravity. The bottom side then grows faster than the top side which causes it to bend upward. As the stem becomes upright again the auxins even out and the stem then grows straight.