There are 3 ways for the U.S. to form treaties:
1- The United States Constitution grants power to the President to make treaties with the “advice and consent” of two-thirds of the Senate. This is different from normal legislation which requires approval by simple majorities in both the Senate and the House of Representatives.
2- The President also makes international “agreements” through congressional-executive agreements (CEAs) that are ratified with only a majority from both houses of Congress,
3- SOLE- EXECUTIVE AGREEMENTS made by the President alone. Though the Constitution does not expressly provide for any alternative to the Article II treaty procedure, Article I, Section 10 of the Constitution does distinguish between treaties (which states are forbidden to make) and agreements (which states may make with the consent of Congress).
The Supreme Court of the United States has considered congressional-executive and sole-executive agreements to be valid . Also, there are self-executing treaties, which do not require additional legislative action, and non-self-executing treaties which do require the enactment of new laws.
However arrived at these distinctions are NOT to affect the binding status of accords under international law. A SOLE-EXECUTIVE AGREEMENTcan only cover matters within the President’s authority or matters in which Congress has delegated authority to the Presiden i.e. ARMS-CONTROL AGREEMENTS are often ratified by this form of treaty.
"All legislative powers herein granted shall be vested in a Congress of the united States, which shall consist of a Senate and House of Representatives."
Please discuss this "international law" of which you speak. What body or individual drafts, enacts, and proclaims it? Who is the sovereign whose law it is? Who is the sovereign who judges the law, and punishes violators, and by what right does that sovereign rule?