It's possible, I guess. The only states that Lincoln carried with less than a majority were California and Oregon. Add their electoral votes to the states Lincoln lost and throw in the four electoral votes he got from New Jersey, a state he lost to Douglas, and it's still not enough to win.
To win, a Democrat would have to carry the whole South and the Border States and a big state -- New York, Pennsylvania, Ohio, or a combination of Indiana and Illinois. You could say that the party split hurt Democratic morale to the point where many Northerners who usually went Democrat voted for Lincoln or stayed home, giving their states to Lincoln.
But who Could the Democrats have run? Douglas was the strongest candidate in the North but he was hated in the Deep South. The other candidates were weak and unknown. Is it possible that a dark horse, an unknown Democrat could have kept the party united and defeated Lincoln, who was also a little-known figure at the time? It's possible, but a weak candidate might not have been able to unite the party.
There's another possibility. Horatio Seymour, the governor of New York, refused to run. If he had, maybe he could have carried New York and won the election. But he didn't. Maybe he's responsible for the Civil War happening when it did.
There's two wild cards, though. One is the Constitutional Union ticket, the old Whigs. Democratic unity wouldn't have kept them out of the race. They wouldn't have done as well as they did in a four party contest, but could still have taken a state or two from the Democrat.
The other wild card is the Southern fire eaters who already wanted secession. They would have found reasons not to support any Northern Democrat. A candidate that they would support would be unacceptable to many Northern Democrats.
The key words here are “fire eaters”.
They had been around since at least 1850, but were kept under control in 1852 by Franklin Pierce and William King’s people.
Again in 1856 they were stifled by followers of James Buchanan and John Breckenridge.
But in 1860 Fire Eaters ran amuck, out of control, first splitting the Democrat party, then threatening and declaring secession due to Lincoln’s election.
So any scenario which imagines Democrat victory in 1860 must begin with a plan to stiffle tbe Fire Eaters, and that admittedly is a tall order.
I’m only saying they did it in both 1852 and 1856, and so we know it was not impossible — where there’s a will, there can be a way.