This is a good thought question because it makes you think about how light works. When you turn on a flashlight, you are creating a source of photons (see How Light Works for details on photons). The photons leave the flashlight and they immediately start to spread out in a cone-shaped beam. Provided that they don't hit anything, each individual photon travels through space forever. So it is not that the photons "run out of gas" on the way to the moon and stop. What happens instead is that, by the time they reach the moon, the photons have spread out tremendously. So few photons hit your eye at any one time that you cannot detect the flashlight.
So the answer to your question is, "It depends on both the flashlight and on the size of your 'eye'". If the flashlight in question is a little penlight flashlight powered by a couple of AA batteries, and if the eye in question is your naked eye, then the answer is, "no -- you cannot see the flashlight from the moon". The cone of a typical flashlight is gigantic by the time it reaches the moon, and the photons are spread too thinly for your eye to detect. If you were to use a much bigger flashlight (for example, an aircraft search light), or if you were to increase the size of your eye by using a telescope, then it is possible for you to detect the flashlight from the moon.
The other alternative would be to replace the flashlight with a small laser. The cone of divergence of a laser is extremely small compared to a flashlight. For example, this article discusses a laser whose beam is so tightly focused that, by the time the light reaches the moon, it has only diverged into a circle about half a mile (1 km) in diameter! You could easily see tightly focused laser light like that from the moon.
The other alternative is to increase the size of your eye with a telescope. A telescope collects light over a large area with its lens or mirror. This is why people use large telescopes to detect the light from distant stars. Even though the stars are very bright compared to a flashlight, they are also very far away (most stars are many light years away, and one light year equals 10 trillion kilometers or 6 trillion miles). By the time the star's light reaches earth, therefore, the light is very dim.
Astronomers can see light from distant objects even more clearly now because of the Hubble Space Telescope, a powerful instrument that has a great vantage point in space. In fact, it has been said that the Hubble can detect the light from a match on Pluto!
This experiment will help you to see exactly how a flashlight's light diminishes with distance -- it's pretty interesting!
Here are some interesting links: