The US had exactly zero nuclear weapons actually ready to drop as late as early 1947. If they had to, they could have pulled together several bombs but they had far, far fewer nuclear weapons than would be necessary to be decisive.

Here's a nice table that the DOE released on their nuclear stockpile from 1945-1948 (search for "Number of nuclear components" — I am failing to translate it into a nice Reddit table!). At most, by 1948, they might have had 50 bomb components. This doesn't indicate how many were assembled and ready to go — which was not very many. Why? Because they were still using World War II-era nuclear weapons until 1949.

OK, but 50 nuclear weapons is not nothing, right? How many would you need to really take out the USSR easily? Well, the USAF thought it would take at least 123 bombs, ideally 466 bombs.

So what would have happened? Oh, they could have done some damage, probably. But the Soviets would have responded with a massive conventional attack in Europe, and the US wouldn't have been able to do much about that. So we're talking about a big, ugly conventional war in Europe where the Soviets had a lot of conventional numerical advantage, and the US having to try and find ways to leverage its atomic arsenal in a way that would swing things. It would be an unpleasant battle.

As for the US public — there was plenty of distrust for the USSR, but much more enthusiasm for war being over, and not nearly the level of distrust/hatred that would come later in the 1940s and in the early 1950s. It would not have been politically popular.

In the aftermath of World War II, there was little desire to go back into war. More importantly, however, there was extensive confusion as to the exact nature of the threat posed by the Soviets. Rather than go to war, the US pursued a policy of 'containment' against the USSR.

This policy ended up being the end result of a series of circumstantial events that happened to intersect with key individuals, combined with the lack of official post-war foreign policy objectives for the United States. A general lack of understanding of the Soviet Union and its own objectives, combined with the fortuitous timing of individuals such as George Kennan and Dean Acheson, resulted in the rapid adoption of containment policy, and the dramatic shift in the manner by which the United States sought to achieve its own national security objectives worldwide.

George Kennan and Dean Acheson served as the principal actors in the formative years leading up to containment’s adoption as formal policy. Kennan served as containment’s philosophical designer and used his background and reputation as an expert on Soviet matters to push containment into the public vernacular. If Kennan can be described as the architect of containment policy, Acheson is the foreman, chiefly responsible for taking the foundations of Kennan’s conceptualizations and pushing for their construction and implementation into the official framework of American Foreign policy.

In 1946, George Kennan, the United States Charge de Affaires in Moscow, sent an eight thousand-word telegram to the United States Secretary of State. In it, Kennan suggested the United States pursue a “long-term, patient but firm and vigilant containment of Russian expansive tendencies,” as a means for the United States to counter what it viewed as an inherent Russian desire to spread communism beyond its borders. This document is known today as “The Long Telegram,” and formed the accidental foundation for the pursuit of containment as a means to counter the Russians.

Shortly thereafter, Kennan's telegram was released to the public. The publication of his internal U.S. Department of State memorandums in the journal Foreign Affairs marked the first time opinions from United States government officials on the looming Soviet problem were released to the public. Although Kennan’s memorandums were never supposed to be regarded as representative of the administration or indicative of foreign policy objectives, the United States had failed to previously establish formal foreign policies regarding U.S.-Soviet relations. This lack of prior official policies, combined with Kennan’s position as head for the U.S. Department of State’s Policy Planning Staff, led to public belief that the publicized documents represented official United States foreign policy. Thus, what had started out as an opinionated personal telegram, was soon codified into American foreign policy, and eventually gave birth to the Cold War.

Sources:

• Frazier, Robert. “Acheson and the Formulation of the Truman Doctrine.” Journal of Modern Greek Studies, Johns Hopkins University Press 17, no. 2 (1999): 229–251.
• ———. “Kennan,‘Universalism,’ and the Truman Doctrine.” Journal of Cold War Studies 11, no. 2 (2009): 3–34.
• Gaddis, John. Strategies of Containment: A Critical Appraisal of American National Security Policy During the Cold War. Revised and Expanded ed. New York, NY: Oxford University Press, 2005.
• JCS 1769/1. “United States Assistance to Other Countries from the Standpoint of National Security,” April 29, 1947.
• Kennan, George. Memoirs: 1950-1963. Boston, MA: Little, Brown and Company, 1972.
• ———. Telegram: The Charge in the Soviet Union (Kennan) to the Secretary of State, 1946.
• ———. The Sources of Soviet Conduct, 1947.
• ———. “What Is Policy?” National War College, December 18, 1947.