not to disuade further answers, but I've written here about this question, and for a bit more context, this answer talks about the post-Civil War weapons testing program, and what the US Army was looking for in a standard platform.

As a quick response, both muzzle loading rifles and repeaters of all types were used in the American Civil War, and according to combat doctrine of the time, repeating rifles were not necessarily considered superior combat rifles, for a multiplicity of reasons.

Part of that stems from the fact that most of what we traditionally consider to be the "Wild West" happened after the Civil War and lasted into the early 1900s. A lot of the characteristics of the "Wild West" did exist before the war, but with respect to firearms, repeating rifles were almost exclusively a postwar thing. The famous Winchester lever-action rifles weren't developed until just after the Civil War with the Model 1866, and Winchester was developing a new version of the gun every few years. Repeating arms were being designed immediately before and during the Civil War, and many models did end up seeing use in limited numbers, generally by cavalry. These include things like the Henry carbine, Spencer repeater, and Colt revolving rifle. Compared to contemporary armies, the Springfield Model 1861 was generally on par with the standard weapons of the Great Powers in Europe and muzzle-loading rifled muskets weren't obsolete at the outbreak of war. Only Prussia had a breechloader (the Dreyse Needle Rifle) and it had plenty of problems that led to its rejection by many, including the British.

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As far as why the Army never adopted a repeater during the war, there were several reasons, all somewhat tied to logistics. First of all was cost. While I can't seem to find numbers offhand for these specific weapons, repeating arms of that era tended to be a significantly more expensive design to procure than your standard rifled musket or even single-shot breechloader. While not a major issue in smaller numbers, for an army that's rapidly mobilizing and generally trying to field as many men as possible, the cost can rapidly become unreasonable. Then there was ammunition. Cartridge ammunition was in its infancy during the Civil War and still had many kinks to work out. Of the designs immediately available to the US at the time, the vast majority were designed around rimfire cartridges, which were traditionally limited to lower pressures (and thus ranges) due to the design and were significantly more susceptible to misfires. While the more limited commercial production may have limited reliability issues, ramping up production of a new cartridge to equip a wartime army would have inevitably seen a drop in quality (and thus reliability).

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And then we have the logistics of rearming and retraining the army as well. Keep in mind that the majority of arms for the Union were being supplied from Springfield Armory after the one in Harper's Ferry was destroyed, although Rock Island Armory was established during the war. The majority of arms were supplied by a small number of government arsenals tooled up to produce massive numbers of guns and ammunition. There was already a variety of infantry arms in service at the outbreak of war - the Model 1861 rifled musket was supposed to be the standard, but the Model 1855 and even older smoothbore designs like the Model 1842 were common. Due to the massive demand for arms, British Enfield Pattern 1853 muskets were imported as well. So with the Army already struggling to standardize around a single gun, asking for a new rifle with entirely unique ammunition in the middle of a war was going to be next to impossible.

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In the field, there was another concern - supply. Concerns over supply in the field were a major factor holding back repeating weapons well into the end of the 19th Century, with the last vestiges of that thinking (magazine cutoffs) only really going away in the First World War. Essentially, in an era where armies were regularly operating far from railheads, supply of ammunition was much more limited than one might expect. During the Civil War, there's many incidences of soldiers running out of ammunition in the heat of battle, and while the wagon train may have more ammunition, the threat of running out of ammunition on an extended campaign was very real. With a single-shot rifle, panicked or undisciplined fire could be kept brief, but with a magazine, a panic could easily cut through a major chunk of a soldier's ammunition supply. Because of that, there was a resistance towards magazine rifles across the world even when the aforementioned problems weren't relevant anymore. And that's also why, post-war, we see the Army (as well as much of the rest of the world) move to the single-shot breech-loading Trapdoor Springfield rather than a repeater like the Winchester.