Southern Democrats did, yes. Johnson as senator at first joined them, then helped push through symbolic civil rights legislation.
As early as 1866, President Andrew Johnson–a southern Democrat–vetoed the first such bill on constitutional grounds. Again, I think he was right in his constitutional claim; in fact, his veto message sounded like Thomas Jefferson.
For my claim that it was the VRA, not the CRA, that ultimately resolved this issue–and thus that segregation could have been ended perfectly constitutionally, and without elevating federal judges to the position of American censors (in the Roman sense), see Klarman’s _From Jim Crow to Civil Rights_.