I don't know if there is research on this, but if not, this might help.
Natural daylight contains a wide range of light frequencies - hence the rainbow, when it's split up. At different times of day, the light has to travel through a different distance in the atmosphere, which tends to absorb some frequencies, and scatter others. This is why the sky generally looks blue (blue and higher frequencies are scattered), but looks redder when the sun is rising or setting at the start or end of the day (absorption of blue and other higher frequency components).
Life on earth has adapted to this, so there are biological changes triggered by the regular cycle of more, then less, of the higher (" blue") frequencies. In simple terms, these changes include triggering of changes within the brain that hint to come awake, or feel drowsy, by working on the parts of the brain that control general arousal. The body has its own internal clock for these, but it also "syncs" to natural daylight, which helps to keep it on track. The triggering is apparently specific on frequencies at the blue end of the spectrum (not on red or other frequencies, or their relative proportions), as far as I have seen it discussed.
So this is the clinical basis behind the advice to avoid blueish light at, or towards, bedtime. Computer screens are also closely connected to mental activity and stimulation, which is a second reason why they aren't too good when heading for a sleepy state of mind.
Now, computer screens are not lit like natural daylight. They don't have a full spectrum - that's an optical illusion. Instead they usually have LEDs in three colours - red, green and blue - and they make all other colours by lighting these to different extents. Apparently the blue frequencies used to make various colours (including white) are close enough to the blue that's triggery in daylight, to have a similar effect.
At this point I have to start making intelligent guesses. What this means in theory is that yes, a computer or smartphone screen could act as a good lightbox in the way you're thinking, simply because it's capable of producing and avoiding blue light of the kind the body's clock is sensitive to. In fact there are a number of apps on smartphones which modify the displayed colours on the screen, to reduce the blue LED component. They make the screen look warmer and redder, or less brilliant white, but in fact they're doing it by selectively reducing the levels of all blue pixels.
So you could in principle create an app that (say) lit your screen a dim red at 6am, and gradually raised the general brightness and brought in the green and blue pixels over the space of say 15 - 30 minutes to reach full brightness. If the blue frequencies are close enough to those recognised by the body clock, it should work pretty much like a lightbox does.
Whether you want that, or want to leave a computer on overnight, or can figure a way to auto-switch on and auto-log in and run the app a while before waking, is a personal choice, though. Also the type of monitor might matter - some monitors have very low visibility off-axis (when not pointing directly at you), which means you are counting largely on reflected light, that may have a considerable filtering/muting effect. ("TN" displays are often like this, and "IPS" displays often have very wide viewing angles within which colours are faithfully seen). A monitor can also have poor colour range, or very low brightness. So it depends on the monitor.
Also bear in mind the noise itself might wake you up too:)