No it won't. In order to pull you anywhere, that gravity has to overcome the gravity you're exerting yourself, otherwise you're pulling objects towards you instead. And for 99.9% of space, this simply won't happen. Our Sun's own influence is minimal once outside the solar system. The solar system seems large in comparison to the distances we're used to living on Earth, but the space between it and Alpha Centauri is several million times greater than the size of our solar system. And for the vast majority of it, neither star is going to be exerting enough gravity to influence a quanta, let alone something with the mass of a speck of dust. This is in fact what makes space travel hard; it's that big that actually winding up anywhere near the solar system you're aiming for is a pretty amazing feat. Needless to say, this wouldn't be a problem if you could rely on it's gravitational pull to get you there.
Ahem. It's scientific fact that the entire galaxy
, thousands of light-years of it, is orbiting the centre of the galaxy. Our solar system and billions of other stars, due to the gravity emitted by the ol' supermassive black hole. You're saying this has no effect on objects inside the galaxy
Seriously, I'm not going to argue this with you anymore until you correct your severe misunderstanding of astrophysics. Among other things:
1. Space is a vacuum. No effect is negligible in a vacuum. Even the light of an electric torch generates a microscopic amount of thrust.
2. The way we calculate the mass of distant objects is by the gravity they exert. Gravity we can detect from Earth/NEO. Thus, their gravity is not negligible, even from dozens of light-years away.
3. The size of an object makes no difference with regards to its attraction by gravity. Come on, man -- Isaac Newton proved this stuff hundreds of years ago.