I wish I didn't say that. I wish I hadn't said that.

Is the second variant possible?
If so, what is the difference between them?

I wish I hadn't said that : its all about the past
means you regret having said something,you said it and its all in the past, you cant change the fact that you said it,

I wish I didn't say that : its about the future or present, you are expressing that you want this situation in the present (or future) to be different.

"Had" is a subjunctive verb. It implies an "if things were different" kind of context. For example, you said it, but you wish you could undo it. Therefore, you use "had."

Use "did" when it's definite: I did not take out the trash.

Use "had" when things could be different: I had not taken out the trash (notice how you feel the need to use "yet" at the end of this sentence? That's because you have taken out the trash, but when talking about some abstract time in the past, you had not done it YET.).

u should elaborate a little more perhaps would help..

This is a confusing discussion!

Yesterday, I wished I had not said the things I said last week.
Today, I wish I did not say what I said yesterday.
Today, I wish I had not said what I said yesterday.

You know, I am not sure! It seems like it should be this way:
I wish I did not say that.

But if I was speaking I would say:
I wish I hadn't said that.
That is what I think is best, but I am not sure!