From the perspective of the US Constitution, the Paris Accords are most certainly not a treaty. Treaties require a 2/3 vote of approval in the Senate.
The tradition of significant international agreements made deliberately short of "treaties" goes back to FDR/Churchill and the Atlantic Charter, if not further.
The proof is in the pudding, here. This is a giant bat signal that the USA is unable to uphold its international obligations. Preen about the Constitution all you want, but that doesn't change the fact that people are going to take the USA's public commitments with a giant grain of salt now, and this will shift smaller countries' orientation to relatively trustworthy large countries, many of which are actively hostile to American interests.
"At a summit in 2014, NATO reiterated its commitment to the 2% target. Members that fell short at the time promised to meet their obligations by 2024."
But, again, even in the worst light possible, a country not spending enough on military spending to meet some stated target doesn't compare to tossing out treaties based on a whim.