When you perform this experiment on the ISS, air is trapped inside the bottle as you insert the neck of the bottle into the ball of water. As you insert it further, the water pushes on the trapped air and the trapped air pushes back. This pressure is the same in all directions.
Unless there is something to prevent it from moving, the ball of water will be pushed away, because the force on it from the trapped air is greater than the force on it in the opposite direction from the ambient air in the ISS. You could prevent the water from moving by keeping it in a plastic container which is sealed around the neck of the bottle. The trapped air would then stay inside the bottle.
Without any gravity to create a pressure gradient in the water, the air in the bottle has no buoyancy, so it does not float "upwards" into the water and out of the bottle. If you plunged the bottle upwards through the membrane of the same container of water on Earth, the air would float upwards through the water; if you plunged downwards it would float up into the bottle.