It's how many commecial projectors work, albeit with very powerful light sources, and tiny TFT displays.
Most graphic LCD displays are STN, that is 'passive' not active, so the contrast ratio suffers as a result. This would mean that, when a pixel is off, it still allows some light through, and when 'on', does not completely allow it to pass. This also goes for TFT displays, although since they have active circuitry, the pixels can be fully turned on/off (each pixel having a transistor and capacitor assosciated with it) so the contrast ratio is in the oder of 400 - 1000:1, as opposed to 4-10:1 for STN's.
LCD's use polarization, the display will always absorb 50% of the light that hits it. There are clever ways around this, such as polarized light emitters (not easy) and polarized reflective films, which 'reflect' the unwanted polarization, rather than absorb it as the filter on the LCD would. This reflected light is then recycled back into the optical system.
Also note, the displays in projectors are very small. VGA/SVGA resolution, but at most a couple of inches diagonally. The size is not only for a smaller unit but also for easier /smaller optics.
I am sure you *could* do it, but you would need a very bright source, and optics with rather long focal lengths (due to the size of the display), making the system a bit on the big side. I have seen DIY LCD projectors using over head projectors and a modified TFT panel - one with the backlight removed. Making a mini one of those would be a great start
Apologies if I've gone over your head, or too simple, I'm never sure what people know