This question has two different parts. First part is why there is no force or reaction happens when an object is thrown.
The answer is that there is a reaction. When one throws an object it need to act on it with a force (applied for example by the hand). The object will react to this force with his inertia (`m*a` ) which opposes the thrown. Thus the equation action=reaction (in this case `F_("hand") =m*a` ) still holds.
The second part of the question is which is the reaction when the object is in the air and falls without difficulty to the ground. The answer to this question is, of course there is a reaction present also in this case.
The Earth is attracting the object with a force `F_1` and the object is attracting the Earth with the same opposite force `F_2 =-F_1` . The absolute values of these forces are
where `G` is the gravitational constant and `R` the distance between the centers of Earth and object.
The difference in this case is that the object has a very small mass and one can observe a relative high acceleration of it due to this force (the gravitational acceleration `g =F/m = G*M/R^2` ). By contrast the Earth has a very big mass and its acceleration due to the reaction is not observable (`a =G*m/R^2` ). In other words not only the object is falling towards the Earth, but also the Earth is approaching on its own the object. Thus also in this case action=reaction and the third Newton law is valid.