The following argument is sound:
- (Premise) All injustices are righted.
- (Premise) If God does not exist, some injustices are not righted.
- Therefore, God exists.
Of course, an argument can be sound but no good. Recall Plantinga's example, which was basically: "God exists or 2+2=5; but not 2+2=5; therefore, God exists", the logic is impeccable, and all the premises are true, but ordinarily only someone who is already a theist will accept the first premise. Is the present argument like this?
Well, I think (2) is fairly plausible. Think of someone innocent who is murdered. Who is there to right that injustice, unless it be God? Of course, one might posit other supernatural hypotheses than that of the existence of God that would suffice to ensure that injustices are righted, but the other hypotheses just do not strike me as very plausible—finite beings like the Furies aren't likely to be able to know and right all injustices.
Of course, premise (1) is not going to be at all plausible to the typical atheist. However, it may well be that some people have an intuition to the effect of (1). This intuition may be related to intuitions about how everything has a purpose, how bad things come to good, etc. These intuitions are ones that even an atheist can have (I once had an atheist student who had such an intuition about her own life—she wondered if the intuition was compatible with her atheism, and I told her that was something she'd need to figure out herself). So the argument need not always be question-begging.