

The usual premise applies, let's try to be civil.
So, I was bored as heck and for the hell of it, I went and rewatched some of the fighting clips from Winter Soldier, Civil War and BvS, cause tbh, those movies legit contain the best action sequences in superhero movies, imo. And I always thought a fight between these two would be hella interesting, considering they're both "just humans" (in the sense that they're pretty underpowered compared to some of the crazier being they're surrounded by, like Thor, Superman, Hulk, Wonder Woman etc.), and both rely on their physical abilities and tools to get the job done, and they're both beasts when it comes to beating people up good.
For the purpose of a fair fight though, we're gonna take out all the gadgets from Batman, and the shield from Captain America. Only hand to hand combat allowed. Who do you think would win, if any at all?
I think it would be a kind of even fight for a while but eventually Cap would take it.
That guy's absolutely insane. Although in my opinion Batman is more skilled in martial arts, Cap is absolutely no pushover either, and the super soldier serum would totally give him the edge. Practically every single physical attribute is superior in Captain America. He's hella stronger (remember in Civil War, he was stopping a fudging helicopter from taking off with one hand and briefly overpowered Spiderman for a sec - and I don't think anyone will think Batman is as strong as Spidey), faster (first movie, when he just got out of the machine, he was almost matching a speeding car), and more durable (in Winter Soldier, he went down I don't know how many floors from an elevator, and still was fine).
I say that it would initially be an even fight because DCEU Bats has indeed showed off pretty impressive strength feats himself (he made a huge crack on a wall just by tossing a guy against it, wtf), and in terms of pure skill, to me he does seem to edge Cap out. However he can't compete with someone with similar (even if slightly inferior) skill but with way superior sheer physical attributes.