Instead of stopping crimes in progress, a real-world Batman might dedicate himself to intense detective work in order to track down robbers, terrorists, serial killers, etc., after the fact. If he did, he'd almost certainly interfere with regular criminal investigations. Could a Batman be smart enough and well-equipped enough to solve crimes better than the police or FBI? Maybe, if that was all he did. But in that case, he's not Batman, he's Sherlock Holmes. If you want a real-world Batman, you have to consider everything else he'd do.
He'd almost certainly use guns if he went into action. There are just too many disadvantages to charging into action barehanded -- no matter how fast you are, no matter how well-armored you are, and no matter how good a martial artist you are. Which means that sooner or later he'd kill someone, and then the police would be hunting him down as just another vigilante kook. (This is assuming someone didn't just shoot him.)
He would not be able to escape pursuit very well. Outracing idiot Gotham police in the Batmobile only works until the first time he gets tracked by a helicopter, or boxed in by organized forces working together, or gets his tires shot out. In person, it would be worse. Hiding in the shadows doesn't work very well if he's being tracked by dogs or night-vision. Jumping from rooftop to rooftop isn't as easy as it looks -- look how much trouble Tobey Maguire's Spider-Man had during his first few outings, and he had powers!
Forget about swinging on a rope from one tall building to the next. To make a single swing without an hour of preparation, he'd have to fire his grappler hundreds of feet, blindly, and basically hope to hell that whatever it latched onto could support both his weight and that of the rope. Without safety gear, he'd have to trust that his gloved hands could support his entire weight as he swung, and hope that he could make a precise enough landing not to get knocked off the rope on impact (and then plunge to the ground). And how many of those ropes could he carry, anyway? There's no way he could easily get the first one back without another fifteen minutes to disengage the grappler and coil it back up.
And let's not even talk about the old "jump off a rooftop without a rope and hope I can find something to break my fall on the way down" business. Surviving that just one time would be a miracle, and he'd get pretty banged up regardless.
Finally, there's the most essential element of Batman: The costume. An outfit like that is neither as scary, nor as identity-concealing, as the comics make it out to be. Yes, he could wear a costume that concealed his height, skin color, eyes, voice, etc., but it wouldn't be all that convenient to wear or move around in-- much less fight in. And it still wouldn't be that scary. Contrary to popular belief, criminals are not a superstitious and cowardly lot-- at least no more than the rest of us. The only way a real-life Batman would be able to make people afraid of him would be to commit heinous acts-- same as the criminals.
So, I'd have to say "no."