Beyond looking at how trying to collimate the beams would be done, there's also a fundamental optical limit on how angularly narrow the final beam can be due to diffraction effects - though in this case, it turns out to likely not matter.
For an approximate answer, I'll take a close-to-best-case set of assumptions: the beam is coherent and Gaussian, assuming a pixel size of 0.5 mm radius (roughly what you have for a standard ~500 DPI phone screen) for the $1/e^2$ intensity width at the waist, so $w_0=0.5$ mm. The worst angular spread will be for the longest wavelength of light, which will be the red pixels at approximately $\lambda=700$ nm (on an aside, for optimizing the spread, you could resize the size of the pixels so that the red ones are larger, shrinking the others so as the spreads end up similar). Final propagation will be through air, which is close to refractive index $n=1$.
This gives a limit of $\theta=\lambda/\pi w_0\approx 0.03$ degrees to either side of 90 degrees. Even if the details of the situation worsen this by two orders of magnitude, at worst, you'll have a few degrees of spread as your limit, which isn't be an issue for most practical applications. For example, the privacy screen mentioned by Rory Alsop has an angular spread of about three orders of magnitude above this limit.