democratic justification of "justice" is an interesting
problem: almost impossible by itself on one hand, it seems
mostly even undesired (because counterproductive to
"ruling-by-judging"-structures) when carefully looked at:

as soon as a person intends to "be judge" (and goes
all along personal recruitment in courts, with studies and
"qualification" and "evaluation" due to the "career" of a
"public employee" judge),
this person obviously is ready, able and willing to establish
a hierarchy of violence (of "state power") among
participants of each process at court.

this is exactly the same for an elected judge:
his/her income depends on elections rather than on recruitment
departments and their exercising of power, but the
hierarchy situation (i.e. antidemocratic situation)
is the same:

dictatorship is only the most obvious hierarchy --
the enemy of each democratic idea is hierarchy
(and not just dictatorship).

(how to "enforce" laws and court decisions without "power",
i.e. "by force" or by violence ("if necessary"),
with even a clear hierarchy of force and violence???)

in other words:
how could a person (or jury) "find" a verdict (decision etc.)
"upon" = over a "case", another person, plaintiff(s),
accused person(s), disputants, if not by evaluating him(her)self
(or being evaluated by others) "higher" than "the others"
in that process? (cf. hierarchy situation above)
how, if not with readiness to (state) violence ("power"
of state), could such a decision be "enforced"???

democratic problems in this context are:
1. compared to lynch justice and "institutionalized"/ritualized
   legal murder ("death penalty") or other consequences of
   "decisions" and "verdicts":
   be that "good" or "bad", (a polarity which always implies
   hierarchymaking, dependencymaking and destruction of
   democratic structures, often even of democratic thoughts)
   it always remains without democratic justification.
2. especially lynch justice would easily correspond to 
   "local" (or "regional") democracy, where the equation
          democracy = majority   (???)
   is attempted to be applied.
   Hitler came into power by majority elections,
   and the "average" majority of "elites" at that time felt
   perfectly comfortable with what happened:
3. being satisfied with ("good") environments and structures
   does not replace their democratic justification.
4. very understandable, "justice by computers" would be called
   inhuman and inadequate to human rights.
   However, "justice" in its form(s) so (far from history to
   now) excludes itself from democratic justification
5. attention: "the public" -- much as "majority" is
   not at all a "better", or a justification "at all"!
   if thought carefully, "the public" is a fourth or fifth
   power in state(s) rather to be carefully looked at
   than "controlled" by its owners or "the state".
6. moreover: looking up law libraries reveals that most
   constitutions in the world do not even contain an
   explicit equality amendment (or at least statement)
   -- very likely to avoid this democratic problem to be
   rationalized.
7. absence of monetary democratic constructs corresponds very much
   to the absence of one of the (now) most important possibilities
   of divison of power.
8. finally, a society recognizing (individual) human rights
   (i.e. of a human being rather than a church hierarchy,
   for example, oppressing its members again and legally then)
   can only have a society-based democratic justification 
   when ensuring a democratic minimum of existence (and 
   not only of "rights" to be "claimed", needing
   time and money to be claimed) to each individual.