The four causes are a direct consequence of Aristotle's explanation of motion as the reduction of one mode of being (potentiality) to another (actuality). Thus, your question amounts to how motion (change) is understood in modern physics.
(Thomas McLaughlin is an expert on Aristotelian vs. modern physics' understanding of motion.)
The question of projectile motion is a good example of metaphysicians grappling with understanding the causality of motion in Newtonian physics. Is projectile motion as treated by modern physics really motion (i.e., the reduction of potentiality to actuality) requiring a cause, or is it really a static state?
For example, E. Meyerson, in his Identité et Réalité (1908), writes (pp. 132, 134):
the principle of inertia demands that we view motion as a state; if motion is a state, it must maintain itself like every state. … The principle of inertia demands that we view speed as a substance. Now this is an entirely paradoxical concept for the immediate understanding
Cf. Stanley L. Jaki's "The Physicist and the Metaphysician" regarding projectile motion, quid quid movetur ab alio movetur ("whatever is moved is moved by another"), and the Principles of Inertia and Conservation of Energy. For more on a metaphysician's perspective on the principles of modern physics, see Garrigou-Lagrange's letters to Pierre Duhem (PDF pp. 14-29).
Modern physics is a "scientia media
" ("intermediate science"), halfway between the first (physical) and second (mathematical) degrees of abstraction—i.e., modern physics, like ancient astronomy, has "a closer affinity to mathematics, because in its thinking that which is physical is, as it were, material, whereas that which is mathematical is, as it were, formal." (St. Thomas Aquinas's Division and methods of the sciences q. 5 a. 3
ad 6). Now,
By its very nature motion is not in the category of quantity, but it partakes somewhat of the nature of quantity from another source, namely, according as the division of motion derives from either the division of space or the division of the thing subject to motion. So it does not belong to the mathematician to treat of motion, although mathematical principles can be applied to motion. Therefore, inasmuch as the principles of quantity are applied to motion, the natural scientist treats of the division and continuity of motion, as is clear in the Physics. And the measurements of motions are studied in the intermediate sciences between mathematics and natural science [physics]: for instance, in the science of the moved sphere and in astronomy.
(ibid. q. 5 a. 3 ad 5)
This gives the reason for Meyerson's quote above. Modern physics is "formally mathematical," and mathematics does not "treat of motion, although mathematical principles can be applied to motion."
Leibniz discussed questions regarding causality and force; he, unlike Newton, was not afraid to give metaphysical reasons for Newtonian concepts. (cf. Pierre Duhem
's Evolution of Mechanics pp. 16-19f.
Further reading (from here