Captain America is going to put so much pressure on Batman that it's going to be tough for him to even think, here lets see how Frank Miller showed Captain America, since you dig Miller:
This is what you're having Batman face.
Batman can also apply such pressure.
If that was the case, why would Batman murder someone? Not to mention in order to pull off "Batman: Hush" first you'd have to stun Captain America, so how do you plan to accomplish that? The comparison doesn't work since you're using the Superman example here, in which Superman is not only fightning the Ivy control, the Ivy control is being momentarily blinded.
Batman wouldn't murder Captain America. My point is that, while using Metropolis's entire power grid stunned Superman, Captain America wouldn't survive such an attack, because he's not invulnerable like Superman. My point is Captain America's far closer to Batman's level than Superman's level of power.
Once again you can't provide a good enough argument in this debate why Batman wins while others here have given plenty of reason why Captain America wins, this is the flaw in your arguments. Either you need to stop it, or start visiting your local library and read some material. Even visiting a wikipedia or a wikia could provide abit more insight for you.
I am not stating that Batman
will win, rather that he
can win. I will definitely tell you that Captain America can beat Batman, but the reverse is also true. It maybe decided by who hits first and with what. If Batman blinds Captain America with a flash-bang or a smoke grenade and kick him in the head, knocking him out, then Batman win. (And yes, he has knocked out a superhuman with a single kick, Deathstroke, in this case)
If Captain America sees Batman and throws the shield (stunning Batman at the very least), then runs over and repeatedly hits Batman into unconsciousness before he can recover, then Captain America will win.
Also, I did base my analysis on the Wikipedia article, which mentioned nothing like him having an immunity to toxins, for example. While I do know that Captain America doesn't tire like Batman will, the importance of that will vary based on the time the battle takes. If the fight lasts only minutes, then his near-unlimited endurance isn't going to help him or provide much of an advantage. I think you are really underestimating Batman here because of his lack of powers. The man almost broke a window designed to withstand a bazooka with only a crack
while being poisoned.
From the strategic and tactical standpoints, they are equal. Batman has more and a wider variety of weapons than Captain America, including many that Cap has no equivalent of his own, but I won't deny that Cap's shield is not only a great tool, but it is also a multipurpose one. The shield is not only indestructible for an impenetrable defense and good for bashing an enemy, but it can also be thrown at an enemy to do great damage.