Why do we fall asleep when our eyes are closed?
You're confusing the cause with the effect. The effect is closing of the eyes (and maybe some component of relaxing.) The cause is sleepiness. In other words, when you're falling asleep, the eyes close, not the other way around. You can close your eyes for 24 hours straight, and you might sleep longer out of boredom, but you'll still spend most of the time awake. Closing your eyes doesn't cause sleep.

The reason our eyes close when we sleep is to protect the eyes from drying out and insuring the oxygenation of the corneas.

Take for example truck drivers. They need to keep their eyes open or bad things happen, like death and destruction. Yet, as they tire, try as they might, their eyes flutter closed and off to sleep they go.

Driver drowsiness detection with eyelid related parameters by Support Vector Machine