Dale, an experienced contributor to this site, offered a surprising explanation for Newton's postulate actio = reactio: In this answer he argued the "explanation [for equal but opposite forces] is the conservation of momentum".
I spontaneously took issue with that because the momentum of a system is state we measure. The law of its conservation means that it does not change across interactions of its constituents. The state is a result of these interactions, not vice versa. In my understanding we have local, "microscopic" interactions between fields and "matter" or, maybe, upon closer inspection, just between fields; and those interactions are of a symmetrical nature.
After my comment to Dale's answer I realize I should elaborate on the "cause and effect" from the title. When we consider an event like the collision of billiard balls or nuclear fission we have a lot of interaction going on that changes the momentum of the involved "parties". What we observe when we compare the system state S before and S' after this interaction is that the momentum is conserved; the sum of all changes is zero. The momentum in the state S' is a result of all changes that happened. That this momentum is equal to the momentum in the prior S is a consequence of the nature of these interactions which happened between S and S'. The symmetry of the interactions is the cause for the observed effect that the momentum change is zero.
To sum up, the symmetric nature of the interaction leads to certain constraints in the resulting state, most prominently the well-known conservation laws. These laws are emergent properties resulting from the peculiarities of the underlying interactions; nature doesn't "know" about momentum (or energy, or angular momentum etc.), and there is no mechanism that would allow abstract concepts to govern concrete interactions. (Of course, from an "anthropic" point of view these symmetries are essential for a stable universe; if interactions didn't preserve energy or momentum the universe would immediately self-destroy or disperse in runaway processes. But that's not a watchmaker fine-tuning the interactions, it's evolution.)
It's possible that my programming background lets me think too much in terms of state-transition diagrams which do not model nature that well: Obviously, interactions never really stop, and states are never really static. On the other hand, many interactions are fairly transition-like, from defenestrations to pair annihilation, so the model is not entirely off.
And I'm aware that the predictive or perhaps rather conceptual power of the abstract, emergent laws is enormous: The conservation laws as well as the thermodynamic laws have a huge impact on our understanding of the cosmos and help form new theories and gain new insights.
But the question remains: Can one in good faith say "forces oppose each other because of the fundamental axiom that momentum is conserved"1, instead of the other way around? Are the two sentences equivalent?
1 I would distinguish this statement from one I would readily subscribe to: "I would be really surprised if forces were asymmetrical because we are very convinced of the general principle that momentum be conserved across interactions. A counter-example would shatter physics as we know it (and likely explode or implode the universe)."
Because, apart from the anthropic argument, the conservation of momentum is not the reason or cause for anything — it is a consequence.