VGA is an analogue format. As far as I know, unless your XBox 360 has HDMI, the output is analogue only (VGA or component - I think probably composite as well - I use component which my amp converts to HDMI to output to the pj).
Apparently you can do full 1080P over analogue VGA or component *if* your screen will support it. It seems that some screens only support it over digital (DVI or HDMI). I know that the PS3 WipeOut is 1080P but I'm not sure there are any '360 games in 1080P anyway. For example Halo 3 wasn't quite 720P (I think it was 640 lines because those extra lines would have lost them some graphics effects).
I'm considering this monitor or the Samsung SM2333SW for a PC I'm building my mum. Either is 23", which is 3" bigger than her 4:3 main TV CRT downstairs (!) Both seem to have HDCP, DVI and VGA according to the reviews on ebuyer.
The Samsung looks very nice mounted in another thread on this forum.
The LG just shot up £20 *while I was looking at it* on ebuyer, so that's out of the running!