It would be a hard fight for the two of them but in the end Batman would win. Personally if I was Batman I would catch the shield as it is thrown at me, then place a electric probe on it and hurl it back so when Cap catches it it will electrocute him.
Captain America. The fight would last long enough for his not-quite-superhuman endurance to give him the upper hand. Unless it was in a varied environment (so, not some kind of empty field or fighting arena) where Batman could win using his stealth/flight/ ranged attacks. Also hi from airport girl
From the comic they fought until the wore each other out in which they thought that they should make a floody diversion so that they can escape and figure out how make things right. Batman was better at swimming.