No, Lincoln ran on a platform of stopping the spread of slavery to new areas of the country. As new free states were admitted, slave states would no longer hold a veto in the US Senate. The federal government would eventually restrict and eliminate slavery from the country. Maintaining the union would result in the eventual end of slavery.
Southern Democrats clearly understood the implications of Lincoln’s words. Consider this excerpt from a secession resolution from Arkansas:
Resolved, that the platform on the party known as the Black Republican Party contains unconstitutional dogmas, dangerous in their tendency and highly derogatory to the rights of slave states, and among them the insulting, injurious and untruthful enunciation of the right of the African race of their country to social and political equality with the whites.
Here is a similar statement from the resolution from another state:
Texas declared its decision to be “based upon the unnatural feeling of hostility to these Southern States and their beneficent and patriarchal system of African slavery, proclaiming the debasing doctrine of equality of all men, irrespective of race or color—a doctrine at war with nature, in opposition to the experience of mankind, and in violation of the plainest revelations of the Divine Law.”
The Republican Party was founded on the proposition equality of all people before the law. The Democratic Party was founded by those who believed in the property rights of slave owners.