Question: In the usual aircraft configuration, the center of lift is located behind the CG producing negative pitch moment (the horizontal stabilizer counteracts it by producing downforce - positive pitching moment). If the aircraft rolls to one side the vertical component of lift on the wings would decrease, so by that logic the aircraft should pitch up.

Why does it pitch down? What am I missing?

Answer: 
The answer is that the aircraft doesn't actually *pitch down*. When banking (or rolling) into a turn, the angle of attack (or the angle between the center of the wing and the flight path of the aircraft) stays the same. Assuming no correction to maintain altitude is made, the aircraft will continue on the same angle of attack, just banked over by some amount. Due to the bank, the lift produced by the wing is no longer vertical, instead, it is directly perpendicular (at least for our purposes) to the wings. Therefore, if the airplane is in a 30 degree bank, then the lift vector will also be 30 degrees from vertical. Doing some vector analysis, then you can determine that if the angle of attack hasn't changed the new *vertical* component of lift will be about 86% of what it was in level flight. Therefore, the plane would begin losing altitude. This altitude loss is what you see as the airplane pitching down.
But tail cannot produce pitch down moment as well. Now tail has to push agains centrifugal force. But gravity is still pulling nose if the flight is not coordinated so it will go down.