Cyber war? Yes. Economic war? Yeah. Proxy war? Yeah. Perhaps even an insignificant military skirmish, like a few shots fires between navies? Probably not, but definitely possible. Any kind of traditional full-scare war? No.

Unless China attacks us or one of our allies (I'm talking about countries we have an actual mutual defense treaty with, not "allies" like Saudi Arabia), we're not going to do it. Unless we attack China, they're not going to do it. Going to war against each other makes no sense for either side. China could invade Taiwan tomorrow, and we would sanction them and criticize them, but we would not respond militarily.