No, but the US has enforced freedom of navigation and ‘law of the sea’ since the days of the Barbary pirates. Like our modern constitutional law stems from British common law, which stems from Biblical law, the UNCLOS stems from earlier accepted common law that civilized peoples know is a requisite for civilized society.
1. The U.S. enforced freedom of navigation and "law of the sea" for American naval and commercial vessels.
2. The "law of the sea" principles outlined in the common law origins you reference almost certainly only apply to "open seas" that do not lie within the jurisdiction of any nation.
3. Any access to disputed or overlapping territorial waters should be negotiated between and with the nations whose territorial waters are involved.