There is so much emphasis placed on precedent in the US justice system and I don't understand why.
With that line of argument I could also argue anything you do or say within the confines of your house is also public's business because it affects the public ultimately one way or another.
There's a lot of value placed on consistency and predictability both in civil and criminal law. If the law were arbitrary, then there really is no single set of rules that apply equally to everyone. So when a court decides X in a given case, any other parties going to a court in the same jurisdiction with the same set of facts and circumstances expects the case to be decided in a similar way. Of course, that's the perfect world theory and caveats apply.
I would emphasize "any other parties": one of the goals is that if I or if someone who is rich, famous or influential were to be defendants in court cases with similar facts, we would receive similar judgements - who the defendant is shouldn't matter, and paying attention to relevant precedents helps ensure that.
I would also mention that the internet tends to misuse precedent in the same way, well, the internet misuses logical fallacies. Finding a precedent you think is applicable to a court case doesn't mean the verdict is decided. Both sides, assuming they have competent lawyers, will submit numerous precedents they believe favors their side, after all. It is up to the courts to decide whether the precedents are applicable and whether to apply the precedent in any give case; precedents inform the court, they do not replace it.
Because the US justice system inherits from English common law. The moral motivation is that it is unfair to judge equivalent cases differently. This contrasts with civil law systems, used predominantly throughout mainland Europe / ex-Roman empire, which have the moral motivation that explicit codification of the law is the best way to be fair.
Civil Law systems have a doctrine similar to stare decisis with the same logic--fairness in outcomes. It just operates differently because of different views about the structure of government.
I think the main reason that Civil Law systems have voiced opposition to judge-made law is because since their advent (French Revolution onward) Continental systems have emphasized democratic control and thus the supremacy of the legislature, whereas Anglo-American law historically emphasized (or reflected) competing power structures. An early French constitution made it criminal for a judge to "invent" a rule to settle a case; if the law was ambiguous the judge was supposed to refer the case to the legislature. But guess how many times a court declared a law ambiguous and referred it to the legislature? (Point being, Civil Law systems noisily refuted the legitimacy of judge-made law, but the reality was different.)
It's notable that Parliamentary Supremacy didn't completely solidify in Britain until the late 1700s. American law didn't inherit such a notion of unlimited legislative authority. The concept of Substantive Due Process emerges from this environment, as does the far less controversial Judicial Review (from Marbury v Madison).
Also, FWIW, there's a common misconception (even among lawyers) that the Civil Law is somehow more closely related to Roman Law than the Common Law. That's not really true. Roman Law actually had judge-made law and something like stare decisis, though the technical details were quite different. Emperor Justinian attempted to impose reforms, but these were widely opposed and occurred at the tail end of the Roman Empire. In as much as the Civil Law was based on interpretations of the Justinian Code, that doesn't mean it was reflective of actual Roman Law. And in any event, most of the major distinctions between the Civil Law and Common Law emerged long after misinterpretations of Roman Law had permeated everybody's legal thinking.
> American law didn't inherit such a notion of unlimited legislative authority.
I wonder if this is due to US's structure being that of a constitutional republic rather than a democratic republic. I.e. the Supreme Court (which executes the constitution) must be superior to the legislature.
There's no reason that the legislature cannot be the arbiter of constitutionality. This is the case in many other systems, including many Common Law nations with a written constitution but which adhere to the doctrine of Parliamentary Supremacy. At the time of the Revolutionary War there were a diversity of opinions on this, though I think resolving the issue wasn't at the forefront of the debate as other questions about the structure of government (especially Federalism) were considered much more interesting and important.
In fact, the Supreme Court has carved out an exception to judicial review that leave some questions of constitutionality to Congress: https://en.wikipedia.org/wiki/Political_question This is especially relevant given the buzz about impeachment because one would normally think that the definition of "High Crimes and Misdemeanors" is precisely the kind of question our courts were intended to answer.
The U.S. is rather unique in investing constitutional review so thoroughly with the judiciary. Most nations, I believe, either adhere to Parliamentary Supremacy or have a special Constitutional Court even though almost every political system subsequent to the U.S. has a written constitution. Also, U.S. judges have lifetime appointments, whereas most other systems permit the legislature to more easily replace judges (see, e.g. the recent controversy in Poland regarding its Constitutional Tribunal).
The U.S. inherited a very strong normative legal system from Great Britain. The nascent U.S. didn't experience the same turmoil and radicalism as did the French, even though there was a ton of cross-pollination of ideas. Because we had such a strong legal system from the outset I think there was less pressure to put all our eggs into the basket of populist democracy, and were able to leave dormant some difficult questions, both at the state and Federal levels. A few decades later (and culminating in the Civil War) we resolved many of them differently (at least in their technical operation) than Europe did because we had set on our own path so early.
Justice Scalia is often held in esteem for his flavor of Originalism and statutory interpretation, but if you read Scalia's scholarly works he very much believed that U.S. courts (including the Supreme Court) should adhere more closely to a Continental European model, one which circumscribes judicial review to the narrowest possible scope. But that's a normative political philosophy that emerged after the Founding of the U.S. (not to mention on a different continent), which means its absolutely not original in any sense. The 19th century Continental European model clearly emphasizes the supremacy of the legislature as the preeminent democratic institution. But this was not the case in the U.S. Issues of Federalism were of primary concern, and those issues meant that the U.S. was grappling with a more complex definition of "democracy", one that wasn't just about how to allow the population to exercise political autonomy in accordance with moral law[1], but how to allow several distinct populations to exercise and moderate their autonomy in tandem.
[1] Moral law meaning concepts of justice and liberty shaped by Western Christianity (including Greek philosophy and Roman law), the Enlightenment, and the Scientific Revolution.
With that line of argument I could also argue anything you do or say within the confines of your house is also public's business because it affects the public ultimately one way or another.