He cited the portion of the Declaration of Independence that is most associated with slavery, not that portion of it that is most associated with "consent of the governed", which is the more accurate summation of the Document's purpose.
Had the Founders wanted to end slavery they could have easily done so, but if so they never would have created the Union -- that Lincoln saved.
In 1776, every state was a slave owning state. By 1787, the vast majority of them were still slave owning states. The founders could not have possibly ended slavery. It would have required the legislatures in every state to do it, and they weren't going to do it because people had too much money invested in the practice and they would not give it up willingly.
Also Lincoln did not save the Union created in 1776. That Union recognized the right of people to separate from a government they no longer saw as serving their interest. What Lincoln did was create a new Union who's foundational principle is that the Federal Government rules supreme over the states. We've been dealing with the consequence of this change ever since.
It has manifested itself in the form of Abortion, Gay Marriage, Anchor babies, Welfare, banning of prayer in public schools, and other social disasters for the nation, all done against the public will.
You may thank God for it.
I don't think the killing of 750,00 people was God's idea. I think that was Lincoln's idea, though he didn't know at the time he started the war, that it would turn into such bloodshed.