Your screen can't get damaged from the sun at all.
If fact, if you turn around so that the sun is shining directly on your LED screen, and then turn off your backlight entirely, you'll notice that if you hold your laptop at just the right angle with respect to the sun, thene the sunlight will actually reflect off the backlight lamp and you'll be able to read the screen with a little bit of squinting. It's harder to do this on modern LCD screens, but if you can pull it off you'll get amazing battery life: the backlight consumes a lot of your laptop's power.