Hi Chris (holmburgers), it looks like quite a bit of interest in your idea, so sorry I'm going to be another party pooper.
I sort of chased down a similar idea long time ago, so maybe I can save you some time.
First, you won't be able to make an actual contact print of the CRT screen, because the glass faceplate keeps you too far away from the image. (You would end up with the equivalent of an unsharp mask.) One way around this is to use a CRT with a fiber optic faceplate, as some commercial printers once did (search for things like Sienna Mileca, or Gretag Netprinter, etc.)
So once you convince yourself that contact printing won't work, you can move on to the next step, using a lens system to project an image. In essence, you would build an enlarger, using the CRT in place of the negative.
What I did, rather than build an enlarger, was to simply cut down some color paper to the size of a piece of film, then, in the dark, lay it in the back of a camera. Then, took photos of the CRT screen. You'll have to go back in the dark to remove your paper, then develop it, etc. But you'll be able to see the effect of an enlarger without actually building one.
The next thing you'll discover is that any color CRT made in modern times has a really difficult time making a red exposure. What you see on the monitor, visually, looks great, partially because of a specific red phosphor (invented in the 1960s, I think), which is really good at stimulating human vision, but otherwise is spectrally lousy. You'll have to drastically hold back exposure on the green and blue. I would guess somewhere around 150 to 200 cc units each, of magenta and yellow printing filtration. My memory is really fuzzy here, but a camera exposure of around 10 seconds at f/4 is probably getting into ballpark range.
If you have a CRT monitor on your computer, you probably have software to invert an image (that is, it looks like a negative), then lower the conrast, you can probably get a decent image onto your color paper. Then, decide if building the "enlarger" (or use large format camera ala keithwms) is worth it. HAve fun!