It is a reaction force, but not a reaction force to the person pushing it down. Reaction forces always act on different objects, so the fact that they add to zero does not mean the object would like to remain at rest. (The ball pushing back on the person is the reaction force to the person pushing on the ball.) The ball is displacing water — pushing water out of the way. The water is pushing back, with an equal and opposite force.
The action of the ball will be due to the forces acting on it: buoyancy, gravity, and (perhaps) someone pushing on it. If the ball is submerged and then nobody is pushing on it and the buoyancy force is greater than the weight, it will push the ball up. If it is less, then the ball sinks. If it is equal, then you have neutral buoyancy.