Thank you for explaining your reasoning. While I understand your point, I think it is based on faulty reasoning.
The two governments involved were the United States of America and the Confederate States Of America, the USA and the CSA,; there was no such thing as the "FSA" or "Federal States of America". Therefore if one army is called "Confederate", based on the official name of its government, I suppose the other side would therefore be properly called the "United" army.
The "United army" is awkward usage in English, unless you are specifically talking of two armies joining forces like Grant and Buell at Shiloh. The most common usage of the time to refer to the army or soldiers of the United States was "Union".