Laws of physics have the same deductive status as axioms within maths. One cannot prove either within each respective system, but for physics, the first selection principle for what is to count as laws is that they seem to work, in the sense of generating predictions that are confirmed by experience. The second selection principle (among those postulated laws fulfilling the first) would be along the lines of parsimony/simplicity.
How did Newton arrive at his third law? Did he just look at astronomical data and then arrive at his conclusion?
If we already know about conservation of momentum, it's easy. Let two objects be initially at rest. Let the two objects be an isolated system -- they may begin to exert forces on each other, but no forces on them come from outside. Let one object begin to exert force F1 on the other object. Assume that the second object exerts force F2 on the first, which is automatically valid because of its generality -- F2 would have the value 0 if it really doesn't exert one, and a nonzero value if it really does. Conservation of momentum, since the total momentum of the system was initially zero, it has to be finally zero. m1 v1 + m2 v2 = 0. Take the derivative of both sides: m1 a1 + m2 a2 = 0. Define the force on any object to be the derivative of its momentum: F = d(mv)/dt = m dv/dt = ma (Newton's 2nd law) . F1 + F2 = 0. F1 = -F2.
I think it would be more accurate to state that physics laws and theory's are verified to be resonably true (or accurate) by experiment, not proven. I don't know if Newton derived his third law mathematically from his other laws or from observation of a relationship between forces and accelerations of varying objects, or from the fact that in any closed system, all internal forces must cancel.
I'm pretty certain the conservation of momentum law was already established by people like Descartes - maybe even before Newton was born. So all Newton probably did was apply some calculus to that already established law and thus 'discovered' the third law of motion. He only really discoverd a new way of looking at it.