It's nice to give it a name that sounds opposite, and ask why it can't exist, but you have to come up with a physical definition before you'll understand the answer.
Matter stretches space towards it. As far as we know, anti-matter does the same (an experiment to test this for sure is planned, and it's surprisingly difficult to test). So there's nothing that makes anti-gravity, bunching up space around it and repelling matter instead of attracting it. A black hole attracts matter and energy so strongly that not even light can escape it.
A white hole would do what, exactly? Repel matter and energy so strongly that not even light could reach it? Anything that repulsive would, instead of bunching up in one place, repel itself to pieces instantly. It would never form a "hole" of any kind because that would be a localized phenomenon. And as far as science knows, no material exists that makes anti-gravity anyhow. So when you have a physical definition of "white hole" you realize that it wouldn't make any sense.
Great question. For now we do not really know, it exists mathematically as a solution to the equations of general relativity, problem is that the calculations tell you that if a white hole exists, it could not be formed at a certain time ... ie if he is there, he has been there ever since. So it stuck on this point, only since we have no theory of proven quantum gravity can not advance further on the ground.
But if it interests you (and I speak only in the context of quantum gravity loops), to make it very simple in this context black holes and white holes are equivalent. A white hole that disappears in the distant future corresponds to the same solution in the void (outside) as a black hole that appeared far back in the past, both have the same effects on matter.
If you want, since you have no more singularity in the center of the black hole in LQG you wonder what happens once the process of forming a black hole has reached the "center", and this process must continue in the future.
From this we understand that the region of space-time that you obtain corresponds by definition to a white hole. (by temporal symmetry)
In the end you get that your white hole is perhaps a physical solution that would be the quantum space-time region that emerges as a result of the formation of the black hole.
These are recent researches and much remains to be done to prove that the scenario imagined here (nottamentally make the calculation of the amplitude of probability so that this process intervenes before the evaporation of the black hole)