Interestingly, F=ma and Hamilton's stationary action are mutually derivable.
The usual presentation is to show that Hamilton's stationary action implies the newtonian formulation. Interestingly, it is also possible to start with F=ma, and move in all forward steps to Hamilton's stationary action. No additional assumptions are required.
I created a series of interactive diagrams to make Hamilton's stationary action entirely transparent.
http://www.cleonis.nl/physics/phys256/energy_position_equati...
(Of course, quantum mechanics is the overall deeper theory; we assume that classical mechanics holds good because classical mechanics is a limiting case of quantum mechanics; at scales large enough that quantum effects average out the outcome in terms of quantum mechanics converges onto the outcome in terms of classical mechanics.)