The problem is not just gamut - it's also dynamic range.
On screen, there is a huge difference in light intensity between the red and the black. Aim a lightmeter at it and the red is much brighter. This contrast is not reproducible on paper - even if the red were in gamut and you could print such a bright, intense red, you still couldn't reproduce the contrast. This is true with any high-contrast image, but this image relies particularly heavily on this contrast for its impact.
Printing it on a high-gloss, metallic surface would certainly help. If the display allows it, backlit transparency would be a far preferable display medium for this image.