No: Culture and technology will persist
Total regression to "the stone age" would require a massively-expensive conflict that collapses the economies of those involved. But it has to go much farther. Countries have continued to fight wars with devastated economies. This war must destroy all trade and all firms and all capital...and all their associated structure...
AND destroys knowledge management (schools, libraries, etc) including professional knowledge (banking, law, medicine, science, technology) and practical survival/fieldcraft (navigation, calendar, weather, domestication, food preservation)...
AND utterly devastates the social fabric of both societies so deeply that parents won't pass much on what they do know to the next generation...
AND, of course, wipes out most infrastructure (energy, communication, food, libraries [again!]) to prevent survivors from collaborating.
Hmmm. Even Genghis Khan, who had some pretty severe policies, wasn't even close to that devastatingkind of devastation. And the question requires BOTH sides to be devastated.
Also, there's also a basic paradox involved: If all structures, laws, social agreements, infrastructure, and technologies of both civilization are in the process of being methodically destroyed, they cannot maintain the complex, expensive, and expansive organized militaries to actually carry out the policies of further destruction.
Finally, not sure how you would keep the survivors from collaborating to rebuild, say, a bit of electrical power or a water pump or a radio or using double-entry accounting or a calendar. A lot of this knowledge is just so darn useful.