How to get a replacement rear axle for a Giant ATX 2

You need to figure out whether its thread is M10x1 or 3/8"x26tpi. It will be one of the two. Could be either.

The main tools to do it properly are a vernier caliper and a thread pitch gauge.

Your rear derailleur pivot bolt threads are M10x1, so there are various tricky ways of using that to determine the thread. 3/8" axles are right around 9.5mm, so you could take one of the locknuts or cones from the axle and see if you can thread it on. It won't be possible if the axle thread is 3/8x26, but will go right on if it's 10x1.

In theory you can squint at it with a ruler to differentiate between 9.5mm and 10mm. The usual as-measured major thread diameters of the two are about 9.4mm and 9.9 respectively.

Then you need to measure the length, which will almost certainly be around 146mm.

Then order a Wheels Mfg chromoly axle of the specs you've found. For anyone who's already broken an axle, it makes sense to cut to the chase and get a fancy chromoly one to attempt to minimize future issues, which probably won't really work in the long run, despite the superiority of Wheels axles. You'll also need a cone wrench to put it all together and adjust it. Hypothetically you can do this with the freewheel on, but in practice it's silly and you should get a freewheel tool as well to take the freewheel off.

One good reason not to do any of this is that before your axle broke, it was probably bent. While it was bent, your cones and cups had all sorts of opportunity to get messed up from being ridden in that state. Getting a good adjustment will likely be impossible. The drive side cup and cone in particular are likely to be more or less destroyed, another reason to get the freewheel off being so you can visually check the cup. Compounded with the probability of bending or breaking it again, it's better in almost all cases to just switch to a cassette hub wheel, which unfortunately means buying a cassette. I personally would check the dropout alignment too, because it's possible for dropout alignment problems to contribute to broken axle situations.