Socialism is usually defined as a system in which the means of production are owned or controlled by the government, or, as Webster defines it, "by the community as a whole." This arrangement was not accomplished by the New Deal, nor was it intended. However, if by socialism we mean any system in which the government creates and enforces regulatory measures for the economy, then the New Deal did represent at least a rudimentary form of socialism.
A large number of Americans have historically been opposed to governmental "interference" with the economy, at least in the abstract. Even today, the United States is unusual among countries of the Global North for the continued opposition, by politicians and a portion of the public, to a plan for nation-wide health care. Yet, one of the chief achievements of the New Deal was the implementation of Social Security, which is clearly a form of socialism by less restrictive definitions of the term—so is the Federal Deposit Insurance Corporation, through which the federal government insures money deposited in banks. After the stock market crash of 1929, the resulting depression was so severe that, at the very least, some governmental measures were required to stabilize the economy. The involvement of the government in the overall system was minor in comparison with what had occurred in both the Communist and fascist states of Europe at that time, and even in democratic countries such as Great Britain.
In the United States today, socialism is a term often avoided by liberals and progressives because it is often regarded as virtually synonymous with "communism" (though the term is coming back into use). Yet not only the New Deal measures but also Medicare (created 30 years later in the 1960s) are in fact forms of socialism in the broader sense of the term, and they retain widespread public support.