Lincoln did not end the institution of slavery when he was elected president for several reasons. The main one was he feared how the Southern states would respond to getting rid of slavery. Lincoln knew that America's political landscape was at a very fragile stage, as there was constant fighting between Northern and Southern politicians in Congress. He feared that trying to immediately abolish the institution of slavery would result in a breaking up of the United States.
Even though he does not do this immediately, Southern states start to secede as soon as he wins the election. Just three short weeks after Lincoln wins the presidency, South Carolina becomes the first state to secede.