These comments raise a bunch of legal and historical issues. First, only Congress can "declare war." But lawmakers can also authorize major military operations without calling them "war." And presidents can attack and retaliate with relative impunity.
Second, in fact, the Korean conflict was suspended by an armistice, not a peace treaty. And that agreement has numerous provisions for monitoring the cessation of hostilities and no clear provision for resuming the fighting.
62. The Articles and Paragraphs of this Armistice Agreement shall remain in effect until expressly superseded either by mutually acceptable amendments and additions or by provision in an appropriate agreement for a peaceful settlement at a political level between both sides.Third, it's worth recalling that the last time North Korea shot down an American military aircraft, killing 31, President Nixon chose not to respond.
The main point I want to make, however, is that historically the United States has always claimed that its wars were defensive, in response to attacks by others, and that Congress was merely declaring that a war exists because of those attacks. North Korea seems to be doing the same.
Here are some excerpts from previous declarations of war:
1812: That war be and is hereby declared to exist between the United Kingdom of Great Britain and Ireland and the dependencies thereof, and the United States of America and their territories;
1846: Whereas, by the act of the Republic of Mexico, a state of war exists between that Government and the United States
1898: That war be, and the same is hereby, declared to exist, and has existed since the twenty-first day of April, A.D. 1898, including said day, between the United States of America and the Kingdom of Spain.
1917: That the state of war between the United States and the Imperial German Government, which has thus been thrust upon the United States, is hereby formally declared;
1941: That the state of war between the United States and the Imperial Government of Japan which has thus been thrust upon the United States is hereby formally declared;