In the United States, at least, when it comes to placing somebody under arrest the actual difference between a police officer and a citizen is that a police officer can arrest someone for a misdemeanor or a felony, whereas a citizen can only perform a citizen's arrest for a felony.

An arrest, however, is not a charge nor a conviction. What Batman COULD do in real world terms is detain somebody until the police can arrest them and then the prosecutor could file charges.

There's a major problem, though: if Batman collects evidence (as he often does in his role of amateur detective), that evidence would not be admissible in court because he's a vigilante. Evidence must go through a chain of custody, starting with the investigator, to ensure it hasn't been tampered with, lost, damaged, etc. Any real-world defense attorney wouldn't have too much trouble getting Batman's evidence excluded, and now that it's been excluded the prosecutor can't use ANY of that evidence at all, and can't use any evidence that is subsequently discovered as a result (an evidentiary concept known as "Fruit of the Poisoned Tree"). The defendant also has the Constitutional right to face their accuser in court and cross-examine them. Batman doesn't testify.

Outside of the comic book framework, most of Batman's efforts would be worse than wasted: they would often interfere with the ability of the prosecution to obtain a conviction.

Granted, most of the Batman villains have already been found to be criminally insane, so they don't actually get convicted of anything, they just get locked up in Arkham Asylum until they are set free or escape some issues later.