This is a very common issue with iPhone 4. The button assembly inside the phone consists of two metal plates, a layer of tape and a tiny plastic nub that's glued to the tape. This plastic nub has either been worn down och has fallen off. The best thing to do is to buy the full button assembly withe sensors and everything, but... DON'T replace the entire sensor because there is a 90% risk that the proximity sensor stops working. It's extremely sensitive. What you do is take a small knife and replace the small plates with tape and everything. It's a bit tricky but I promise, it's the best sollution. Take the phone apart and remove the screen so you easily can unscrew the button assembly.
A stupid question maybe, but did you remove the protective film on the back of the new display? The most common cause of this issue is that the replacement glass is of poorer quality and a bit softer than the original, so it flexes when pressurized. This causes the back of the LED-display to get squeezed between the glass and the metal plate housing.
It seems to be a bug in iOS6 on 4S. I've had the same problem and for me the sollution was to connect the phone to iTunes and when iTunes has found the phone you take out and insert the sim-card again. It might take a few tries. Try that and report if it worked or not.