Responsibility for those deaths lies with those who first started, then formally declared war on the United States, on May 6, 1861.
That would be the Slave Power which ruled the Confederacy.
You are confused or poorly informed about the history of the war between the states.
Neither side ever formally declared war on the other.
The secessionist southern states did not declare war on the United States.
The southern states declared their secession from the Union based on their belief that states, which had voluntarily joined the union, had retained the right to separate from the union.
The federal government did not agree that the states had the right to secede from the union and declared the secession to be a rebellion.
The government determined to put the rebellion down by military force rather than resolve their disputes through diplomacy.
But he did not. In fact, he rejected overtures from the south to negotiate.
If Abe Lincoln was the great statesman and president many believe him to be, he would have pursued ways to solve the dispute without the loss of 600,000 killed, 500,000 wounded and the almost complete destruction of the south.
The 600,000 dead was 2% of the entire US population at the time. That would be equivalent to 6,000,000 dead today.
But the war dead and injured are not the full story.
Lincoln approved of U.S. Grant's tactics of loosing the union army on the civilian population of the south to punish the secessionists. Grant openly stated his determination to destroy the south and Lincoln acquiesced. General Sherman was sent out to burn the south to the ground and that is what he did.
The Union army's brutal raping, killing, pillaging, plundering of the southern civilian population, the destruction of the railroads and other infrastructure, and the burning of cities, homes, farms and businesses, even after the south surrendered, is all well documented in the history books.