Can a pizza stone be fixed after soap has been used to clean it?

I would rinse well with clear water. Allow to dry thoroughly, and then use as normal.

There is a small chance that your pizza stone will impart a soap flavor to your pizza, but I would say that it is worth a couple of pizza cooks to determine if there is a long term problem. It's really not that much of a risk.

By rinsing, you can only remove stuff from the surface and slightly below. You need to remove molecules which sit in the pores of the stone.

Chemically, you are working against diffusion and adhesion. By washing with soap, you have deposited a number of molecules onto the stone surface, these have diffused into the numerous pores of the solid. Now, these molecules cling there by means of adhesion, which makes it really hard to get rid of them.

Thus, instead of just rinsing, let it sit in a bowl covered with warm to hot water. Heat and time will allow the molecules to slowly "come out of the cover" and go into solution instead. This is driven by the concentration of soap in the solution (you want little to none) and temperature (you want it hot!). Preheat the stone if you can.

Add a drop of oil to the liquid, and disperse by beating with a fork (or anything comparable - the idea is to have as many small droplets as possible). This will act as a trap for the soap molecules in the water and reduce the solute concentration - increasing the pressure for the remaining adhering molecules in the pores to go into solution. If possible, try beating foam, and if some foam forms, then remove that and introduce new oil.

I suggest somewhere around at least 30-40 degrees Celsius and a timeframe of 10-30 mins.

These are just guesses. Also, depending on what material the stone is made of, there might be additional reactions and/or interactions involved that could require a higher temperature or longer times.

Rinse it thoroughly and just cook some dough on it to throw away (instead of a full pizza with all ingredients). I don't believe the soap will be that resilient to withstand rinse+heat+food on it. It is not designed for that.

If you have a self cleaning oven, run the pizza stone though a cleaning cycle in the oven. The oven will heat up slowly enough to not cause thermal stress in heating. The oven locks for hours to allow for a long cool down cycle to avoid thermal stress when cooling. The stone was manufactured at much higher temperatures than you'll get in an oven.

Then I'd rinse the stone, and air dry it.

You might try giving it a good coating of oil. That could either help work the soap out or "bury" it in the stone.

My wife washes off our pizza stone from time to time in sudsy water. I really freaked out the first time I saw her do it, but since then I've calmed down and we've used it many times (and she's washed it a few more times) and I never could taste any soap in the pizza or whatever we were cooking.

TL;DR = It's no big deal. Don't worry about it.