Comets and stars are two very different things.

Part 1 - Comets

Comets are quite close to us (for space!) but pretty small in size. They're part of our solar system, and if the only things that were in the solar system were the Earth and comets, one of three things would happen to each comet:

- If it was moving fast enough, it would fly off into deep space. The earth's gravity would slow it down a little as it went but not enough to stop it escaping

- If was moving just at the right speed and in just the right direction, it could be captured by the Earth and go into orbit around it. This is a pretty unlikely scenario though, so it would only happen to a few comets

- If it wasn't fast enough to escape, and not on the right path to go into orbit, it would eventually be pulled right down to Earth, just like you suggest. "Eventually" could be a very long time, because when the comet is far away, the pull of Earth's gravity is very weak. But the solar system is over four billion years old, and there would be time for that to happen eventually, if not already.

However, the Earth and comets aren't the only thing in the solar system. The Sun is by far the most massive thing in the system, with more than 300,000 times the mass of Earth. That means its gravitational pull on any object is also more than 300,000 times bigger than the pull from the Earth.

So basically everything that I said would happen with comets and the Earth happens, but with comets and the Sun. The Earth pulls comets too, but it's gravity is puny, almost nothing, compared to the Sun.

So after 4 billion plus years of the solar system, all the comets have either escaped into deep space, fallen into the Sun or another large body, or gone into orbit around the Sun.

Part 2 - Stars

Now stars are a whole different story. They (like the Sun, which is the nearest star) are huge, and much more massive than the Earth. But they are also incredibly far away.

a) The Sun

Before we get into the distant stars, it's interesting to think about the nearest one - the Sun. Would it be stupid to ask why the Sun doesn't fall down to Earth? Actually no!

Gravity is a two way force. The Earth pulls on the Sun with exactly the same force as the Sun pulls on the Earth. It obviously makes sense to wonder why the Earth doesn't fall into the Sun. But for a physicist it makes even more sense to wonder why when the two things attract each other, they don't "fall on each other", by coming together.

The answer is the same as we saw with the comets, the Earth's speed and direction of travel is such it orbits the sun rather than falling in.

Now if the force is two way like I said, why does the Earth orbit the sun, and not vice versa? The real answer is they both orbit a point in between their two centers of gravity! But the Sun is so massive that that point ends up inside the Sun itself. But two obejcts that are near in size can end up orbitting round a point in the middle, and that actually happens quite often when two stars are close together.

b) Stars in our galaxy

The other stars we see in the night sky are immensely far away compared to comets, the Sun, or other planets of solar system. They are pulled by the Earth's gravity, but that is very very weak at that distance. Just like the Earth's gravity doesn't have much effect on a comet compared to the Sun, it has little effect on the stars compared to other forces acting on them. The stars are in orbit too, but they are in orbit round the center of the galaxy.

c) Everything!

Ok, so if everything atttracts everything else through the pull of gravity, why has it not all come together yet? (Which would really be like ALL the stars falling on the Earth!)

Given enough time, that would happen, but for two rather important things.

- Imagine if on the surface of the Earth every person was pulled towards every other. What would happen? Would we all end up bunched together in a huge ball? Not necessarily. Because while some of the forces pull you East, others pull you West. While some pull North, other pulls South. If people are spread evenly all over the Earth, the net effect would be that you're not pulled in any particular direction. This is basically what happens in the universe on a large scale, gravity pulling one way is cancelled out by gravity pulling the other way.

- Now imagine the same thing, but the Earth is expanding, blowing up like a balloon. Far from all the people being pulled into one huge clump like we were imagining, they'd actually be getting further apart. This is what we think is happening in the universe - the space of the universe itself is expanding! At the moment we really can't tell if it'll go on expanding forever, or not. That's part of what people study with the Hubble telescope and such.

So finally...

It turns out your simple question about why stars and comets don't fall on the Earth turns out not to be so simple at all, in fact quite difficult and very interesting.

It reminds me what's so mind-boggling and fun about science and especially gravity.
Source(s):
Data on sun's mass vs earth's here:

http://www38.wolframalpha.com/input/?i=sun+earth

And here's a comet picked at random:

http://www65.wolframalpha.com/input/?i=C%2F1989%20Q1%20(Okazaki-Levy-Rudenk...

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