There was a serious coup involved in the American Civil War. It was a self-coup by the legislatures of the various Southern states. (For those who’ve never heard the term self-coup before, it’s probably the most common type of coup and the textbook example is something like “I, the lawfully elected President, am going to suspend elections and declare myself President For Life”).

First, to get the obvious out of the way, Secession was not a consequence of anything Abraham Lincoln actually did as President. A simple look at the Timeline of events leading to the American Civil War shows that clearly. Abraham Lincoln was only inaugurated President on March 4. By this point South Carolina, Mississippi, Florida, Alabama, Georgia, Louisiana, and Texas had all seceded and formed the Confederacy. None of them in response to any actions taken by Abraham Lincoln other than winning an election.

After the Confederacy formed it does not matter if secession was legitimate or not, what Lincoln did was neither a coup nor illegal.

If secession was illegal he was justified in using military force to bring the armed rebellion to an end. Neither a coup nor illegal. If secession was actually legal he was justified in making war on the hostile state on the southern border of the United States that had attacked the United States first (at Fort Sumter). Neither a coup nor illegal. Either way Lincoln was morally and legally justified and he handled things appropriately. Making war on a hostile state that attacked you is not a coup if you have the backing of the legislature.

However. There was the self-coup I mentioned. Andrew Weill and Rod Young have both brought up different parts about how the initial document was the Articles of Confederation and Perpetual Union and the Constitution talks about forming a More Perfect Union. Secession was illegal.

But more to the point there wasn’t an actual mandate for secession by the State Legislatures. To give them their due, the Texans were aware of this and did hold a popular vote over secession (but this was also a case of a self-coup with the legislature ignoring the legal powers of Governor Sam Houston who was against secession because he could see what was going to happen and was trying to not call the legislature and to veto what he could).

North Carolina held a popular vote over whether there should be a secession convention - by a majority of 651 voters on February 28 1861 they voted not to even have a convention on whether to secede. The North Carolina legislature had the militia seize United States property. Then the legislature was admitted to the Confederacy even before the secession convention the voters of North Carolina had voted against. The legislature decided not to submit secession to a vote of the people despite that being a part of secession.

In Tennessee the voters voted against calling a secession convention. Which didn’t mean the legislature didn’t secede and join the Confederacy. In this case in the vote taken after the secession and while at war with the United States the voters opted by a large majority to endorse the decision.

The only state I’ve found where there wasn’t a self-coup was Arkansas (as pointed out by Mark Ferguson in the comments) where the secession convention originally voted to stay in the Union with a popular vote to be taken in August. After Fort Sumter they reconvened and voted to secede.