In short, no, the Nazis did not invent the term.

"Politically correct" as the concept we understand it today, i.e. as a pejorative towards discriminatory language or as a pejorative to describe efforts to achieve a less discriminatory language in certain milieus has only recently (in the 90s) entered German discourse. In fact, it has also only entered recently into American discourse mainly due to its usage in the NYT in the early 90s.

The term has been used to describe the Nazis though. One if its first usages as a pejorative term comes from an article of the NYT on December 31, 1934 describing Nazi German efforts to curtail journalists. In that context it was used to denote the Propaganda Ministries effort to impose a language regulation on journalists in the Third Reich. In fact, what scattered reference we have to the term being used is in this context, also e.g. referring to the Soviet Union like a NYT article from 1949 referring to the newspaper "The Daily Worker" as politically correct.

The end of the Cold War was what shifted the meaning of the term more into the direction we use it today. The Nazis had virtually nothing to do with it rather than being described with it by a now probably discarded meaning of the term.

Sources:

• Geoffrey Hughes (2009), Political Correctness: A History of Semantics and Culture.

• Irving Lewis Allen: Earlier Uses of Politically (In)correct. In: American Speech, Vol. 70, No. 1 (Spring, 1995), pp. 110-112.