Make sure your floating gameobject has a Rigidbody component attached to it and set 'Is Kinematic' to true (make sure checkbox is checked). When your button is pressed, use the action "Set Is Kinematic" inside Physics drop-down to enable gravity on the gameobject. It will fall using physics...
NOTE: The floating gameobject must have a collider to interact with other objects or it'll pass through everything and continue falling until oblivion...