The best conclusion one can reach based on the evidence is probably April of 1861, specifically April 12-15.

South Carolina was the first state to secede (December 20, 1860) in response to Lincoln's victory in the presidential election, and six more states (Georgia, Florida, Alabama, Mississippi, Louisiana, and Texas) seceded between that date and February of 1861. Bear in mind, this was before Inauguration Day had been moved to January 20, so James Buchanan was still in the White House this entire time. Bucahanan, a "Doughface" Democrat from Pennsylvania, was not at all enthusiastic about states leaving the Union, but he also did not believe in compelling them to remain in either. Throughout the Secession Winter, there were multiple attempts to reach some kind of arrangement to prevent a shooting war and bring these states back in. These included the Crittenden Compromise and the Corwin Amendment, which were proposed schemes to amend the Constitution to entrench protections for slavery in the Constitution at the federal level. Keep in mind, between abolitionist activist movements in the North, the John Brown Raid on Harpers Ferry in 1859, and the North's resistance to complying with fugitive slave laws, there really wasn't a whole lot of willingness on the part of secessionists to believe unionists were at all serious about following through on this proposed obligation. It is also important to remember that northerners had just for the first time elected to the presidency a candidate from an avowedly anti-slavery party (the Republicans) without a single southern electoral vote. Secessionists knew that politics in the North did not favor much in the way of compromise. Likewise, antislavery Republicans who had just one an election on a platform of opposing slavery in the western territories couldn't be seen to cave in to demands from the slave states.

By Lincoln's inauguration on March 4, 1861, seven states had left the Union, and eight more slave states that had yet to make a decision (Virginia, North Carolina, Tennessee, Arkansas, Kentucky, Missouri, Maryland, and Delaware). These were states that bordered free states, or where slavery didn't play quite the same socioeconomic role that it did in the cotton belt. These are also states that had some of the largest white populations needed for military manpower of any of the slave states, and these states also have some of the South's best available industrial infrastructure, Virginia being a prime example with the Tredegar Iron Works in Richmond, the federal armory at Harpers Ferry, and the Wytheville lead mines (lead is kind of an important resource for military purposes). If the Confederate States of America was to survive as a coherent, viable nation-state, it was going to be easier with these states inside the tent, so to speak. These were states that had political classes that were not enthusiastic about Lincoln's program, but were also very skeptical of the more extreme brand of southern nationalism espoused in places like South Carolina and Mississippi. These states were critically important. There's an apocryphal statement attributed to Lincoln that "I would like to have god on my side, but I must have Kentucky." Even if he never actually said it, the basic principle was true: these states were geographically strategic based on population, resources, and geography. It was not entirely clear which way these states were going to go.

That changed in April of 1861. One of the outstanding issues between the CSA government and the USA government was the occupation of federal forts in the newly seceded southern states. The Confederate states considered these forts to be their sovereign territory, but Lincoln refused to withdraw US forces. The most important of these installations ended up being Ft. Sumter, which sits near the mouth of Charleston harbor. I'll gloss over the blow-by-blow surrounding everything that happened, but to keep it short, Lincoln made the decision to resupply the fort, as it was being besieged by Confederate forces. At this point, Confederate authorities made the decision to take the fort by force, and the bombardment began in the wee hours of the morning on April 12, 1861.

This is really the point where a full-on shooting war wasn't perhaps inevitable, but extremely difficult to avoid. Historians don't generally deal explicitly in counterfactuals, so we can't say with specificity what would have happened had Lincoln backed down in this situation. What we do know happened, though is that on April 15, President Lincoln issued a call for volunteers to suppress the rebellion in the seceded states. This call went out to all of the states, including the slave states that had not yet seceded. It was at this point, in response to Lincoln's decision to call up troops to bring these states back into the Union, the Virginia, North Carolina, Tennessee, and Arkansas seceded in April-May 1861. Maryland, Delaware, Kentucky, and Missouri all remained in the Union, and many of Virginia's counties constituted themselves as the new state of West Virginia in the middle of the war. There are some troop movements and small engagements in these first few months, but you really have to wait until July 1861 with the First Battle of Manassas/Bull Run for the large scale engagements to begin.

Any answer to a question is admittedly arbitrary, but I think mid-April 1861 is the best answer to your question, as this is the point when the shooting started, and when the political actors who could have backed down chose not to.

Readings

This is a topic that gets written about a lot, but I can recommend a few things.

For general Civil War history, and history that includes background and context preceding the war, it's hard to beat McPherson's Battle Cry of Freedom.

For something a little more focused on the topic at hand, William Freehling's The Road to Disunion, Volume II: Secessionists Triumphant, 1854 - 1861 focuses more specifically on why the Civil War broke out when it did. It's a bit denser and more academic than McPherson, but it's still a good read.

If you want to get really focused, I can recommend Lincoln and the Decision for War: The Northern Response to Secession by Russell McClintock, which is a more microscopic look at the specific subject you're asking about.