False.
Southern Democrats supported slavery, or rather, viewed it as as an issue best settled by individual states, whereas Northern Democrats opposed its expansion into new territories and also supported a national goverment. The Southern Democrats ended up cleaving off from the Northern Democrats in 1860, having their own candidate. So the Democrat Party had two separate candidates running. The North, btw, being more populous, means the majority of Democrats were not pro slavery, or at least not in favor of expanding it.
As seen with the Lincoln quote, this was in line with his attitude, as well. He allowed the Border States to retain the practice of slavery, exempting them and captured territory from his bogus Emancipation Proclamation.
Further, round the 1880s, the Lily-White Movement faction of Republicans emerged in the South, seeking to exclude 'Blacks' from their ranks and favoring segregation.
So, no, as history has shown, slavery, segregation, Jim Crow, etc, wasn't so much a party issue but a Southern one. That the lingering beliefs of such is embraced now by mainly Republicans in the South is further testimony of this.
Also, for those not knowing, the Confederacy permitted no political parties, thus it being a Democrat vehicle is a ridiculous lie.