A couple with a notable size difference: a photomontage of the Earth and the Moon
No, even under the most adverse circumstances that is not something to be reckoned with. It is true that two bodies with large masses, such as the Earth and the Moon, exert a strong gravitational pull on each other. If this were the only active force, the two celestial bodies would indeed crash into each other. However, the centrifugal (outward) force of the Moon's motion counteracts this gravitational pull. The Moon moves through space at a very high speed, and its inertia would cause it to move away from the Earth were it not for the fact that the gravitational pull forces it into an orbit around the Earth. This equilibrium of forces could be disturbed, however.
One could image several different catastrophe scenarios
One could for instance imagine that air resistance would slow the Moon down. Our Moon revolves around the Earth in an almost circular orbit at an average distance of 384 400 kilometres, so more than 30 times the Earth's diameter. This far away there is no trace of the Earth's atmosphere, which means that there is no air friction either.
Asteroids would also not be able to knock the Moon out of its orbit and onto a collision course with the Earth. An asteroid would have to be much more than 1000 kilometres in diameter to be able to accomplish this. However, the largest asteroids in the vicinity of the Earth, such as Eros, only have a diameter of 30 kilometres at the most. Ceres, the largest asteroid in our solar system, with its diameter of about 1000 kilometres, is in a stable orbit around the Sun between the planets Mars and Jupiter - so it will not get close to the Earth. In any case, the Moon would most likely be destroyed in such a collision.
The Moon’s orbit around the Earth is not stable
The tides (ebb and flood) on Earth, which are caused by the Moon, do have an important effect, though: the tidal friction slows down the Earth's rotation. The Earth and the Moon form a system in which the total energy of their rotation and revolution (the 'total angular momentum' which combines the rotation of the two bodies and their revolution around their common centre of mass) stays constant. As the Earth's rotation is slowing down, the kinetic energy of Earth-Moon orbit increases - the Moon very slowly spirals outward, away from the Earth. Using lasers, scientists were able to establish that the distance between the Earth and the Moon increases by about 3.8 centimetres every year. This means that we could lose the Moon a few billion years from now.