Why isn't the capillary tube displaced downward by the column of water?

We do. The walls of the tube are pulled down. However, virtually in all meaningful situations, the tube is sufficiently anchored such that this small force is undetectable. It just gets transferred onto the ground/table/etc.

However, were you to do this experiment in zero-gravity, and expose an open tube to a drop of water, you would indeed see that the water moves one way, and the tube moves the other way. The end result would be that the center of mass of the water/tube did not change.

Hypothetically, if you put the tube on some hyper-sensitive crush washer which was calibrated to crush just over the force the empty tube applies, you would find that adding water crushes the washer. The water would be pulled up, and its mass would be supported by the tube, and thus supported by the washer.