I know quite well, as you probably do, that international law has zero bearing on the military decisions made by the USA in terms of its interests. Yes, it'd be illegal, unless the UN Security Council gave its blessing. Just like the second Gulf War was illegal. That has never stopped the US before.

> How does it correspond with "we are the nation of laws"?

It does not. The United States government rejects the idea it is restricted by international law on these matters.

As for South Korea, it can argue it's an internal domestic matter.