To understand why it won't work, you need to know how a laser actually works.
In this picture, you can see the 3 main components- the mirrors, the gain medium (ruby) and the pump (the flashlamp). To understand how a laser works, you need to understand how these three components come together.
In the simplest form of the laser, a flashlamp is used to pump a gain medium, which causes the atoms in the gain medium to enter a raised energy state. To return to the ground state, these atoms emit radiation in the form of a 694nm photon (visible red light). As more of the atoms enter this higher energy state, a population inversion occurs (where there are more atoms in the raised energy state than in the ground state). Eventually, some atoms will begin to spontaneously release these photons. These photons may collide with other atoms, causing them to release their stored energy as a photon. These waves of photons resonate between the mirrors, eventually escaping through the partially silvered mirror.
In practicality, lasers are much more complex, and that was just a watered-down, back-to-basics explanation that works. It's a common misnomer that lasers use radiation, instead, the radiation being referred to here is actually light radiation, with the whole acronym being Light Amplification by the Stimulated Emission of Radiation.
That aside, let's take a look at an argon laser. In this case, the gain medium is the argon gas, and the pump is the electrical current flowing through the tube.
The reason why argon works as a lasing medium is because it emits, and does so quite easily. You may be able to get other substances (such as water even) into an excited state, but there's no point, as it won't emit photons.
With Radon, it doesn't have a useful emission band, and it can't be made to readily emit at this band, so it'd be pretty useless even if it weren't radioactive. Combined with the fact that it is radioactive only intensifies the problem. Dealing with a laser tube that has an effective life of a few days, and which settles into radioactive components is not fun.
Radon has bands in the 700nm range. There's easier ways to get 700nm or so laser light, without dealing with the short half-life, radioactivity and the difficulty in exciting radon. Ti:Sapphire, for example, is tunable from 690nm up to 1000nm, covering those bands.