Other answers are right at explaining why tidal forces move Earth and Moon apart but they don't move apart a pair of black holes. However, I think it is also needed to explain why the phenomenons making two black holes spiral inward don't make the Moon spiral inward to the Earth.
In fact, every pair of rotating masses radiate gravitational waves. What makes the difference is that only very large masses rotating very close to each other produce gravitational waves large enough to meaningfully affect those masses orbits.
According to https://en.wikipedia.org/wiki/Gravitational_wave#Binaries the time that takes a pair of masses to fall into each other due to radiated gravitational waves is:
$$t= \frac{5}{256}\, \frac{c^5}{G^3}\, \frac{r^4}{(m_1m_2)(m_1+m_2)}$$
Let's plug the masses of Earth and Moon and its distance into that equation (all data taken from Wikipedia in SI units):
> G <- 6.674e-11
> r <- 384e6
> mluna <- 7.342e22
> c <- 299792458
> mterra <- 5.97237e24
> (t <- 5/256*c^5/G^3*r^4/(mterra*mluna)/(mterra+mluna))
[1] 1.304925e+33
That is, left alone, radiating gravitational waves would make the Moon crash into the Earth in 1.3*10^33 seconds, that is 4.13*10^25 years or 3*10^15 times the current age of the universe. In other words, the effect of radiating gravitational waves in the motion of Earth and Moon is so tiny - specially compared with other forces like tide ones - that we can't absolutely forget about it.
Just for comparison, two one solar mass neutron stars orbiting each other at the same distance of the Earth and the Moon would fall into each other in:
> msol <- 1.9885e30
> (t <- 5/256*c^5/G^3*r^4/(msol*msol)/(msol+msol))
[1] 2.19985e+14
Which is just about 7 million years, showing that changing masses have a large effect on the outcome. As stated in the beginning, gravitational waves make pairs of star sized objects to spiral inwards but they don't have noticeable effects on a satellite orbiting a planet.