The president of the US does not have the power to start a war without getting it approved by the UN security council. You're arguing internal implementation details, but the legality is not determined by your courts.
The US constitution specifically calls out treaties signed by the US (such as the UN Charta) as supreme law of the land. Article VI, the "Supremacy Clause".
Thus, US law, too, defers to international law.
Please at least read the legal framework you're so confidently misdescribing.
A law defines the nature of collective action in response to certain violations. Words on paper themselves are impotent. If there is no potential for enforcement, i.e. there is no counterfactual state of collective action, there is no law.
It's been the established president since the Korean War when the US began ignoring the constitutional provision that gave congress the power to declare war. Additional examples are the Vietnam War, Serbia, Afghanistan, Iraq I & II, the Libyan regime change, and the current Iran conflict, and there are plenty more. The written law still states that the president does not have this power, but the actual unwritten law has been that he can. And that is the only law that matters.
Yes, the US and not just the presidency. If it was just the presidency then he would have been impeached by congress for usurping their constitutional authority.
The war is certainly illegal. Our systems are just so atrophied at this point that we treat congressional approval as a formality. This is a choice we make over and over again that we need to stop making.
US constitution says that starting a war must be authorized by Congress, president has no authority to do it on his own.
The problem is: over time the US grew so powerful, that the definition of "war" became blurry. "No, we are not at war, our soldiers are just dropping bombs on Iran for fun and profit".
EDIT: Another problem, of course, is that current member of Congress have no balls to stand up to Trump and reclaim their constitutional powers.
Congress made its mistake a long time ago. Power is very difficult to reclaim once it has been relinquished. And it didn't even take a Caesar crossing the Rubicon in our case.