However, the withdrawal didn't cede any particular power to the President in that regard--it merely said that the United States or the USSR could withdraw on six month's notice.
Left up in the air is the specific mechanism of withdrawal, and there the clause that says that all treaties and laws made pursuant to the United States Constitution are laws, with an implication that they are subject to legislative and executive action in order to repeal them.
However, the United States Senate did not argue the point. Silence implies consent.