There's two parts to your question, I think: why does The Doctor always appeal to authority, and why does he appeal to that authority so frequently?
The Doctor's Behavior
The Doctor doesn't really assume that everyone abides by the law; he is giving them the chance to abide by the law. One of the few consistent aspects of his personality is that he really would prefer to let "bad people" correct their own behavior before he has to step in. We see this from the very first revival episode all the way through the confrontation with the Silence. He calls out his opponents behavior for several concurrent reasons:
- Give then notice that he's detected their bad behavior and knows that it's a crime, as @Keith points out.
- Make it clear that he has a greater awareness of the larger universe than the victims (who are often humans), with the unstated implication that he's more capable of fighting back.
- Give them notice that he, personally, is planning to try and correct their behavior if they don't correct it first.
It's possible that he's hoping to "scare" the enemy into running away, the way Eleven did in his first episode or Ten did with the Vashta Nerada, without him having to actually do anything. I doubt he actually expects the Shadow Proclamation to arrive, or plans to "turn them in". I suspect he's already resigned himself to having to take action, but his own personal moral code means he has to give them a chance to do the right thing first.
The Shadow Proclamation
However, the biggest question you're asking, I think, is why that authority should make any difference to anyone.
We don't really know much about them, just bits and pieces. There's more information in related media but I haven't read much since the revival. The TARDIS Wikia lists a bunch of off-screen mentions of them, including an IDW comic where Ten is put on trial (meaning he apparently does recognize their authority as legitimate).
The Shadow Proclamation appears to be the closest thing in the universe to a globally established police force. There's some sense that they "replaced" the Time Lords after they disappeared but I don't know if that's "canon". We do know that the first time we hear of them is from Nine, who also mentions later on that then the Time Lords were around they acted as a kind of universal police force. It seems likely that The Shadow Proclamation popped up to fill in that void.
They were around at least as early as 1 BC -- the possessed priestess in Pompeii knew of them. We know the Judoon work for them, and they appear to have both the technology and motivation to act as a global police force. Other races (like that Adipose lady) do appear scared of them. They appear to have highly advanced technology, lots of historical knowledge (one of the few races that knew about Time Lords in any detail), and pretty solid information gathering when we see them in Stolen Earth.
Where they get their authority isn't as clear, but it seems like the same place any police force does. If enough people simply accept the Shadow Proclamation's right to enforce their rules by force, that makes them de facto authorized to do so because no one's going to stop them. It's entirely possible that they are just strong and scary enough that any space-faring or time-faring races knows of them and is scared of them, and that might be exactly what The Doctor is hoping for.