Yes. A whole lot of it, in the sense that most of it is time pressured, and frequently allows for just a single check in the midst of something else happening before someone else does something. We might not call it a "round" or a "turn" because nobody else other than the fighter could actually do two things in that time frame. But once you introduce a fighter with action surge, provided the player is thinking about how to use it, absolutely the time sensitive challenges out of combat can use it.
Does a lot of your skills and spells used out of combat to resolve challenges happen with no time constraints? Are your PCs not doing things which require stealth and speed, or making checks to influence a conversation mid-conversation?