Would Lincoln have had support for a war if he hadn't provoked the South into firing first?
By sending a letter, he "provoked" the south into firing on a fort that was built on a sand bar that the State of South Carolina ceded to the Union in 1805 along with Ft Moultrie and a couple others. In 1836, South Carolina officially ceded all "right, title and, claim" to the site of Fort Sumter to the United States, as the construction of the fort was done on federal dollars and with 70 tons of New England stone for it's foundation stone..
When SC attempted independence, she thought that the forts she had given the Union and the one the Union built and paid for were somehow hers.
Not very gentlemanly of those southern gentlemen of the day to try and steal something that they willingly gave away previously or that someone else built and paid for..
Lincoln told the Governor of SC that a civilian ship carrying food, water, fodder would be sent south and a week later, the south committed the first act of it's suicide.