In short term, Dr. King’s strategy did very little to undermine the culture of Jim Crow and segregation in the South. However, it did a great deal to undermine that culture in the long term.
In the short term, nonviolent civil disobedience did not change the minds of people in the South to any great degree. That is why the nonviolent protests of King and his followers were met with such extreme violence in many places. If King’s tactics had worked to change the culture of the South, segregationists such as George Wallace would not have been as successful as they were. After all, Wallace was able to win five Southern states running on a third party ticket in the year that King was assassinated.
However, King’s strategy did change things in the long term, largely because the strategy worked in the short term on Northern whites. Whites in the North supported King to a great enough extent that laws like the Civil Rights Act of 1964 were able to pass. With the passage of such laws, the country changed dramatically. This led to the erosion of the Jim Crow culture in the South over the long term.