Mitt Romney has a lot of nerve portraying himself as a civil rights leader. He and his church have been on the wrong side of three major civil rights battles in the past 40 years.
1) Racism: The LDS Church was openly, blatantly, officially racist until 15 years after the Civil Rights Act. They taught that blacks were less humans, descended from Cain and less valiant spirits in Heaven. They taught that the American Indians and Hispanics were descendants of apostate Jews, the Lamanites. They removed Indians from their families to place with white families to improve their lives. In 1989, they excommunicated George Lee for calling them anti-Indian. There is still a lot of unofficial racism in the church today, although they tacitly acknowledged that Indians are not corrupted Jews by recently changing Joseph Smith's preface to the Book of Mormon.
2) Sexism: The LDS Church proudly helped defeat the ERA. They still limit women's leadership roles to women and children's organizatons. A woman's highest ranking in the church is "Sister", and the most prominent women are spouses of leaders. Women were taught not to work outside the home until fairly recently, and are given inequal status as missionaries. A man is still head of the home in Mormon eyes.
3) Homophobia: Mormons are among the most homophobic organizations in the US. They have given millions to fight gay marriage laws, and still excommunicate church members for being gay. Romney himself is the George Wallace of gay marriage. In 2002, when running for governor, he promised MA voters that he was a supporter of gay rights. In 2004, he did everything in his power to prevent gay marriage while he was governor, including begging the US Congress to amend the US Constitution. He used 1920s miscegenation laws to limit non-resident gay couples from marrying, an irony in itself. His antigay stance dropped his popularity to the low 30s and lead him not to seek a 2nd term in MA.
For Romney to claim stake any claim to King's legacy ignores the long history of regressive stances that he and his church have taken over the last 40 years.