The "United army" however is awkward usage in English, unless you are specifically talking of two armies joining forces like Grant and Buell did at Shiloh, then in that case it also is merely only another adjective. The most common usage of the time to refer to the army or soldiers of the United States during this period was "Union" because it reflected the NAME of the government.
Federal is only an adjective like Yankee or Rebel, Northern or Secesh, it is in no way reflective of the title of the actual government that raised the army, and gave the orders, and controlled its use.
The CSA and the USA were both systems of a "federal" government, (as men in Georgia and Texas State militias found to their distaste ) so the term "federal" could actually and equally apply just as validly to troops of the CSA, since they were also a federal government.
The troops raised in Georgia and Texas and elsewhere in the Confederacy became, against very stong objections by many, part of their own "federally" controlled army. The CSA asserted their own federal power over "states rights" through the draft and implied threats of force. The CSA considered itself a "higher authority" over the individual states. In the CSA the war power of their federal system trumped the power of any state individually and the draft was the perfect case in point, as well as the power the CSA invoked to send Georgian troops into combat outside of Georgia.
Let's put it this way, from the point of view of a citizen of Georgia during the War, the CSA was their "federal" government. It tells the exact relationship of any state to the higher more powerful central authority.
A Yankee sympathizer today uses the term "federal" for the same reason, it politically recognizes the "higher authority" of the central government over any individual state, it is simply a political term about political relationships between authoritized governments of unequal status.
So for any Rebel sympathizer today to call the Army of the United States THE "federal" army, is to give tacit recognition, that it was THE army of a "higher authority" of a central government that officially did not recognize the CSA as a legitimate entity, and was thus, by its point of view, dealing only with rebellious states over which it held authority. Calling that army the "federal" army implies the recognition of that relationship. Using the term federal in this context, for the person who uses it, implies recognition of that status.
Both CSA and USA were federal systems. On the other hand, both groups of states, in the North or the South, were "confederations" of states. Both were also American. So how best to distinguish them in a neutral manner?
To distinguish between the two governments and the two armies, the easiest and best choice is to focus on the offical TITLE of the governments.
So, it all comes down to official title of the governments involved. The CSA's army is called Confederate because of the TITLE of the government. The USA's army is called Union because of the TITLE of its government, with the word "United" modified for grammatical reasons.
The army of Great Britain isn't called the "Britain army" because it makes for awkward usage in the English language. So for the same reason the USA's army of 1861-1865 is not called the "United army" for gramatical reasons. "Union" is more grammatically correct.
After 1865 these was only one army for the one government, and no need to distinquish it from any another army operating on US soil, therefore today we commonly refer to the army of the USA as the AMERICAN army.