Why do you want to clear the screen? It's a debugging window.
Closing it, and reopening it will clear it. It will also restart the Arduino.
You could also write your own serial window application. That's what I did. I have a Find field, to find data in the window. I can add additional features are they occur to me.
Or better yet, instead of using the IDE serial monitor, use a real serial terminal program. If you are using Windows I highly recommend Brey terminal, it has tons of features, it’s free, it’s a standalone exe file, so no installation required. Along with tons of over commands, it has a clear receive window, clear send window.
This will not work in the Arduino Serial Monitor, so you need a terminal emulator. Since it doesn't use a serial port, you also need a serial port emulator. DOSBox and Procomm will work. Both are free.
Download and install DOSBox (I am using v 0.74), then modify the config file under [serial] and set serial1=directserial realport:COM1 (or whatever COM port you are using).
In Procomm, set the appropriate baud rate, using alt-p, and no parity, 8 data bits, and 1 stop bit.