Why doesn't viewing blue sky hurt us like viewing blue light of digital screens? It is said that all digital devices these days emit blue light rather than UV; long term exposure can hurt our eyes. Why doesn't viewing the blue sky for a long time hurt us? Is there any big difference between these two blue lights?
 A: Blue light inside earths atmosphere
The daytime sky receives its color due to Rayleigh scattering, as more blue light is scattered:

It is for the scattering, that the light that hits earth, thus, what you see, has less intense blue light. And thus is a little less harmful.
Experiments
To actually tell if a laptop monitors emissions are more harmful to your eyes than the blue sky, you will need to perform a few measurements and calculations:
You will need to measure the power (per area, at the distance you usually have from the screen) radiated by your screen, which depends on the settings, the currently displayed frame (its colors) and the type of screen you use. An example LCD display for instance has the following spectrum:

You would need to also measure the power (per area) of the skys blue light on the ground to compare them.
Generally I'd say there is no reason to perform an experiment with your own eyes though.
Concluding
The damage your eyes experience depends on the time they are actually exposed to the specific light. I always find myself looking away from the blue skies after a few seconds as it usually feels too bright... and look at my PCs monitor for hours.
