Its well known that the U.S. is a common law country, but I'm not sure if that's codified anywhere (like the U.S. constitution) or if just happened organically because that's the way its always been. Is there anywhere officially establishing common law in the U.S., or is common law a precedent in and of itself?
Generally a precident. The original 13 states were all using English Common law prior to the the Revolution, and as a good number of the people who ran the United States were used to this system it pretty much just was. Most states that were added to the U.S. after the 13 were either territories formed from existing states, or territories that established and decided to adopt common law as law of the land. In so far as I can find, only two states, Louisiana and Califorina, had to change systems in some way. LA uses civil law for third party procedures and retains common law for just about everything else. California retained the California Code from it's Spanish Law origins, but is fully Common Law. California Code laws are subject to Common law judicial changes.
To say that the United States is a "common law" country is a descriptive term, rather than one that is mandated by a statute.
But, the U.S. Constitution is interpreted in light of the legal understanding and meaning of words of the founders as of 1789 (the body of the U.S. Constitution) or 1791 (when the Bill of Rights was adopted) or at the time that later amendments were adopted.
For example, when the courts determine what is meant by a jury trial under the 6th and 7th Amendments to the United States Constitution respectively, they interpret that in light of what a "jury trial" meant as of 1791, and incorporate in 7th Amendment jurisprudence the distinction between cases at law (where there was a right to a jury trial at common law) and cases in equity (where there was not a right to a jury trial in 1791 and which the courts have determined that the founders did not intend to create one). So, the issue of whether or not you get a jury trial in federal court (and in most state courts) hinges on case precedents in England and the U.S. colonies from 1791 and earlier.
Many states have statutes that incorporate by reference into the common law precedents of their state courts, all common law precedents of England which were in effect as of July 4, 1776, subject to later revision by the courts of that state in accordance with principles of stare decisis or as subsequently limited or changed by state statutes or federal law.