The black hole thing's a lot different than the media suggests it to be. Assuming a perfectly spherical planet, the gravitational force on an object not in the planet is G*m*M/r^2, where G is a constant, M is the mass of the planet, m is the mass of the object, and r is the distance between the object and the centre of the planet. (This is technically a certain limit of General Relativity, which seems to be more "correct," but I won't go into that.)
So, if the Earth shrunk to the size of a black hole without changing mass and remaining spherical, the gravitational force on the Moon would remain constant; there wouldn't be any gravitational effects. Someone who was standing on the surface wouldn't have any ground underneath them, but otherwise, the gravitational force remains constant.
Now here's the clincher. For small objects (say, a black hole,) you can get closer to the centre before having to enter the object. And, as r gets small, G*m*M/r^2 grows ... pretty freakin' huge.
So it's not a problem of density, but rather of distance.