What would happen to the moon if earth stopped exerting the force of gravity on it?

At the instant when gravity turned off, the moon would leave its curved orbit,
and would sail off in a straight line, in the direction it was going when gravity
disappeared.

Not really.  When you look at the bigger picture, it turns out that the moon
is much more in orbit around the sun than it is around the Earth.  Its orbit
is very nearly a solar orbit, but with some relatively shallow dimples in it
that result from the gravitational influence of the Earth.  If that influence
suddenly disappeared, the moon would continue in its solar orbit ... moving
either slightly faster or slightly slower than the Earth, depending on whether
it was nearer or farther from the sun when Earth's gravity disappeared. 
We would see the moon either slowly pull away from us toward the west,
or slowly fall behind us to the east.  Either way, it would get smaller and
smaller and eventually disappear in the distance, and it would come around
to meet us again roughly once each year.