The short answer is that nobody knows, and serious work on the problem seems just to be getting started. A minimum figure would be around 50,000 -- this is the number given by James MacPherson, based on the impressions of contemporary witnesses. Modern statisticians know that such estimates are almost always too low, and that major wars produce "excess deaths" far beyond what people around battlegrounds see and count.
But how to carry out statistical tests? A quick check of the census shows that the population of the southern states grew by 11% between 1860 and 1870. So in that sense there are no "missing" people. Of course the population of the US was growing by leaps and bounds in the nineteenth century, and between 1850 and 1860 the population of the southern states grew by 24%. But lots of things might have depressed population growth: immigration to the South was cut off during the war by the Federal blockade and very low for years afterwards; births fell because so many men were at war or away from home working in factories; many blacks left the south for the north, and many whites headed for the west; economic dislocation, of which there was plenty in the south, usually leads to falling birth rates. So you can't simply compare the growth rates and blame war deaths for the missing 13%. Not only that, but the 1870 census was later attacked as the most unreliable in US history, and it may have missed as many as a million people in the South, 8% of the population. If so, that would make up for much of the shortfall.
In contrast to the number of civilian deaths, the number of military deaths has gotten a lot of statistical attention. In 1866 the government estimated 260,000 dead Union soldiers and 100,000 Confederates. But those estimates slowly rose, and by the end of the 19th century the government was able to count around 620,000 dead men, 360,000 in the North and 260,000 in the South. But that much disputed 1870 census seemed to show far more "missing" men, roughly 500,000 in the North and 350,000 in the South. These higher estimates are getting renewed interest from historians, I suppose because statistical studies of contemporary wars in Iraq and other places have documented so many more war deaths than anyone was able to count. Plus, modern studies seem to show that the 1870 census was not that much worse than any other census -- perhaps a 6.5% undercount, compared to compared to 6.0% in 1850, 5.5% in 1860, and 3.6% in 1880.
I find it interesting that from 1866 on the assumption has been that most of the "missing" men, North and South, were soldiers who were killed in the war or died from the lingering effects of their wounds. Nobody seems to be suggesting that a lot of white male civilians somehow died. But comparing these numbers gives the maximum figure for Southern white male civilian deaths as 90,000, the difference between the 260,000 documented dead soldiers and the missing 350,000 men in the census. Also, all of these calculations include comparisons between the male and female populations which assume that the women are all still there. If anybody had thought that tens of thousands of civilian women had died in the war years, wouldn't somebody have mentioned it? Southerners did a lot of complaining about Northern behavior during the war, but I am not familiar with any accusations of murder on a genocidal scale. Surely there were excess deaths due to hunger and the diseases spread by roving armies, but not so many that contemporary Southerners made this a big part of their bill of Northern perfidy.
Another thing that seems clear is that in the south, blacks suffered more from the war than whites. Many lost or fled from their homes, and whenever there is a food shortage the poorest people suffer the most. Many of the contemporary accounts of civilian deaths specifically focus on southern blacks, and this is part of the reason (not just abolition politics) that northern aid societies and the government launched programs to help freedmen.
It seems to me, going over these numbers, that the number of Southern white civilians who died almost has to be less than 200,000, and that a reasonable estimate might be 50,000, well under one percent of the population. Which qualifies as a tragedy but hardly as a genocide.