Space is very big. Is it actually likely that said waste spreading would even be noticeable?
They also thought the Pacific Ocean was very big and that the waste wouldn't even be noticeable.
Car drivers still think this way with their toxic emissions. It is very human to think 'out of sight out of mind' rather than 'take only photos, leave only footprints', so maybe it is actually fine to trash space with waste.
The only case in which releasing waste into space is problematic is when it's within a highly-utilized area, for example LEO. In the future, we may potentially use various transfer orbits and/or the Interplanetary Transport Network [1], so we should presumably avoid dumping there, but even within the solar system, the sheer volume of vacuum dwarfs any possible use.
[1] https://en.wikipedia.org/wiki/Interplanetary_Transport_Netwo...
Let's call it $20,000 / lb to orbit (http://www.businessinsider.com/spacex-rocket-cargo-price-by-...) and say double that for the "disappear into the distance" part.
That gives us $160B to loft the US's nuclear waste for one year into not-my-problem-ness.
That doesn't sound unreasonable, right?
P.S. Don't drop it.
Is the unreasonable portion, not the cost.
There is risk, but by coal is even more dangerous. The risks are all easy to mitigate if you use modern engineering.
A more likely scenario is that the Earth's orbit will move outwards as the Sun expands, since stars in the red giant phase(s) lose a lot of mass. It's plausible that the Earth/Moon will not be destroyed, physically. But in that case all life would have been obliterated due to the immense temperatures associated with having the surface of the Sun within a few million miles (or less) of Earth.
Nuclear explosions as a method of space propulsion is a viable idea [1], however. (Using the Moon would be rather inefficient though.)
[1] https://en.wikipedia.org/wiki/Project_Orion_(nuclear_propuls...