After audio surfaced of Representative Jason Lewis, a swing district Republican from Minnesota, making misogynistic remarks on his former radio program, his Democratic opponent in November’s midterm elections is saying Mr. Lewis’s rhetoric proves he is too extreme to represent the ideologically diverse district.

Angie Craig, a former journalist and health care manufacturing executive who is challenging Mr. Lewis in one of the most competitive districts in the country, said in an interview that “comments like this don’t represent Minnesota values.”

Mr. Lewis’s remarks were first reported Wednesday by CNN, which unearthed audio recordings of a radio show he hosted from 2009 to 2014.

In several instances, according to the recordings, he implied that women who voted for Democrats proved the stereotype that the female gender is more governed by emotion than reason. He also said that women who vote based on issues like birth control “have no cognitive function whatsoever” and lamented that certain women can no longer be publicly called derogatory terms like “sluts.”