The United States had a standing army before, during, and after The Civil War (although expanded during the war).Over 97% of those who served in The Civil War were state troops under federal command.For a lot of reasons it was important that the states kept track of "their" men thus a lot more records for state units (made by the states).
The US Army was only the concern of the federal government(not quite that simple but close enough)and a record set I doubt you want to go to without having a lot more information and likely not then.
The bottom line is that Rev. RH Carter would be MUCH easier to search for if he had been in the 5th. Ohio Infantry rather than 5th US Infantry(assuming he did serve in this unit).
In both cases it is likely to be little genealogical information until you get to pension records.