It supplies power, but not much at all.
The answer is no, you can not.
Can I tap a small screen's 12V power to the ignition wire in my radio harness so that it will turn on when my ignition is on?
You can use this 12V to control a relay.
it will sure work like crinos said, but in my car i just attached the screen to this wire, works perfectly. A screen doesn't use much power, so probably won't be trouble. A friend of mine did it the same way, but that's the same car (same brand, same type). Maybe other cars will have trouble with it !
true, my civic has the same, now that you mention it.