To be sure the US did not have to recognize the CSA, there was no law forbidding them to leave the Union. This right was fought for and paid in blood at Cowpens, Yorktown and many other battlefields.
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Servia, Land of Serfs:
“What should this new [Southern] nation be called? Since there were questions of more importance to be settled in Montgomery, in a matter of fact way the constitution makers called it the Confederate States of America. Yet there were suggestions that it be called the Republic of the Southern United States of America, and Thomas R.R. Cobb wanted to call it the Republic of Washington. As time went on sundry other names were suggested, such as Appalachia, Alleghenia, Chicora, Panola, or even just Southland.
The Federals liked to call it Secessia, which did not displease the Richmond Whig editor too much, for he felt that the United States might well be renamed Servia, as it was a land of serfs made so by Lincoln’s tyrannies.
But this editor and other strongly State-rights Southerners wanted none of these names – not even Confederate States of America, for that indicated a nationality. They hated the word “national” when applied to the South; there was no Southern nation, they argued. There were eleven nations in the South; they hated the word “State,” as it was a Yankee term. They would compromise on “commonwealth”; but the term “League of Nations” should be applied to the whole, or “The Allied Nations” or the Allied Republics.”
As for the people, historically they came to be called Confederates…and though their enemies delighted in calling them “rebels,” the Southerners took up this term very early and gloried in it. They liked to recall that George Washington was the first great American rebel and Martin Luther was another great rebel. In fact, “Southern” was especially disliked by some, as it indicated merely the southern part of the old Union.
(A History of the South, Volume VII, The Confederate States of America, 1861-1865, E. Merton Coulter, LSU Press, 1950, pp. 58-59)