I don't know the answer to this, but

Legally, how is it possible for a private citizen to violate another private citizen's civil rights?
I don't know if this is the section Nelson was charged under in the federal case, but if anyone could explain this, I would appreciate it.

You can argue whether or not the outcome of the state murder trial was justified, that's really not my concern here, since it's a dead letter.  I'm concerned about whether the subsequent civil rights prosecution was legal.  If the state murder acquittal was a miscarriage of justice, it's not the first time.  Look at Issei Sagawa.  Part of having a mature understanding of this sort of thing is to realize that the criminal justice system isn't perfect, and that it is inevitable that innocent people will be convicted, and that guilty people will be acquited, from time to time.  That's the price you pay for having a legal system; in the end someone has to make the decision, and sometimes they may not necessarily get it right.  I don't think it's going out on a limb to say that the Sagawa case is completely beyond all reasonable belief.