A) The Senate ratifies all treaties. Not the president.
B) Congress has the power to pass international laws, not the president. Most of the issues that presidential administrations negotiate over are given to the presidents BY ACT OF CONGRESS.
https://www.law.cornell.edu/constitution/overview
Article I, Section 8 gives several "foreign policy" powers to Congress:
To regulate commerce with foreign nations, and among the several states, and with the Indian tribes;
To establish a uniform rule of naturalization, and uniform laws on the subject of bankruptcies throughout the United States;
To coin money, regulate the value thereof, and of foreign coin, and fix the standard of weights and measures;
To provide for the punishment of counterfeiting the securities and current coin of the United States;
...
To define and punish piracies and felonies committed on the high seas, and offenses against the law of nations;
To declare war, grant letters of marque and reprisal, and make rules concerning captures on land and water;
Article II, Section II on presidential powers:
He shall have power, by and with the advice and consent of the Senate, to make treaties, provided two thirds of the Senators present concur; and he shall nominate, and by and with the advice and consent of the Senate, shall appoint ambassadors, other public ministers and consuls, judges of the Supreme Court, and all other officers of the United States, whose appointments are not herein otherwise provided for, and which shall be established by law: but the Congress may by law vest the appointment of such inferior officers, as they think proper, in the President alone, in the courts of law, or in the heads of departments.
The current negotiations with Iran are largely over sanctions, which are imposed by the executive under laws passed by Congress.
The letter may have been unwise, but it surely wasn't unconstitutional.
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