But why? What do the two sides have to fight over? What territories do they contest? Much like the Cold War, this is made for a proxy war where you do have two sides face off over territory or resources with the two sides supplying the proxies. Maybe a fight over Taiwan, but I don't see the US really stepping up to defend Taiwan unless they get the rest of the world to join them in a coalition. And right now we got Mr. No Friends Necessary burning bridges with every coalition we had.
When you get away from the rhetoric I see little reason for the two sides to have an armed fight. Economic battles and trying to buy off allies very well could happen, but nothing that suggests armies engaging.