I don't think you're going to find a single word for this particular emotion. Words for emotions in English Wikipedia list are most closely tied to the how you're feeling: happy, sad, angry, depressed. You're looking for a word that's a combination of the how (satisfied/happy) and the why (justice has been served).
As such, I think you're going to have to settle for using an adjective or adjective phrase. Some adjectives you could consider are:
vicarious. Merriam-Webster.com. 2014. Encyclopedia Britannica Co. http://www.merriam-webster.com/dictionary/vicarious (accessed: July 21, 2014).
experienced or felt by watching, hearing about, or reading about someone else rather than by doing something yourself
- Karmic - adjective form of Karma, which you seem to be familiar with
As others point out in comments, your particular situation has the added wrinkle in that the source of the satisfaction is ambiguous. Are you feeling satisfied that the victims have been compensated? Or are you feeling satisfied that wrongdoers have been punished? "Receiving justice" after all can be applied equally to the plaintiff and the defendant. Or are you primarily satisfied by seeing the systems itself working?
All of these could trigger a different or better noun to combine with your adjective of choice.
Regarding your comment, since you sway away from any sense of revenge, I'll go out on a limb and propose you're mainly interested in words describing fairness. As such, the most concise phrase that I can think of that describes your scenario exactly is: vicarious, equitable satisfaction.
Vicarious is covered above and elsewhere. Synonyms include: proxy, surrogate, indirect, borrowed, derivate (also derived, derivational), second-hand, provisional, vicarial (especially if religious sentiments are included), sympathetic (buy with an implication of siding with the plaintiff), considerate.
Equitable means characterized by equity or fairness.
noun, plural eq·ui·ties.
1. the quality of being fair or impartial; fairness; impartiality: the equity of Solomon.
Synonyms: disinterest, equitableness, impartiality, fair-mindedness, fairness, justness, evenhandedness, objectivity; justice, probity. Antonyms: bias, discrimination, inequity, injustice, partiality, partisanship, prejudice, unfairness, unreasonableness; injustice.
2. something that is fair and just: the equities of our criminal-justice system.
equity. Dictionary.com. Dictionary.com Unabridged. Random House, Inc. http://dictionary.reference.com/browse/equity (accessed: July 21, 2014).
The latin root here is very interesting. Equites were originally horsemen or calvary in the
Roman military and were later distinguished as a social class mostly for their citizenship. Equities were eligible to serve on juries, and from there the meanings of fairness and justice were added to the word. Much of modern governance, including the democratic electorate and justice systems, are based on Latin roots as well. In particular, the Maxims of Equity have been a framework for many legal systems and include this gem:
Equity delights to do justice and not by halves.
If you want a term that is less steeped in legal tradition, you could use karmic, cosmic or a suggestion one of the other answers in place of equitable. A discussion of the differences between karma and justice would probably take up as many words as I've already put into this answer. I think it's sufficient for us to know that there are nuances of meaning between them.
Finally, satisfaction is the closest emotion to the scenario you describe, which means:
2 a : fulfillment of a need or want
b : the quality or state of being satisfied : contentment
c : a source or means of enjoyment : gratification
Which captures the pleased feeling you describe. However, satisfaction is also a noun for the compensation that the plaintiffs receive, which you expressly state you are not among.
3 a : compensation for a loss or injury : atonement, restitution
b : the discharge of a legal obligation or claim
satisfaction. Merriam-Webster.com. 2014. Encyclopedia Britannica Co. http://www.merriam-webster.com/dictionary/satisfaction (accessed: July 21, 2014).
That is why, together with the vicarious & synonyms, it best describes your situation.