Research I've done on a former New York City ferryboat, converted hastily to become the gunboat USS Commodore Barney, causes me to agree with the comments of Harve202. At the start of the American Civil War, the Union Navy consisted of only 41 assorted vessels, so a concerted, rush effort was made to purchase a widely assorted group of civilian craft for blockade and river patrol duty. In many cases conversion work was done in a matter of weeks with no time for such such finishing touches as paint.
In correspondence, the captain of one former ferry notes that another ferry remained white, the color it was painted when in civilian ferry service. It would appear that ship's carpenters and crew members were frequently called upon to make minor structural changes to many vessels while they were on station. Thus, it would not be illogical that crew members used any available or captured paint on their craft. What does appear to be clear, is that there was no standard color scheme on the relatively few that were painted. Weathered wood, soot and rust dominated.
A likely exception to this among Union Navy vessels might have been the gunboats that were built during the war. They had a better chance of being painted; almost always with gray of varying shades, depending more upon availability than choice.
I'm not familiar enough with the Confederate naval forces to be sure, but due to the even greater scarcity of materials, I'd guess that the same situation prevailed for the South. The exception would have been the sleek, foreign built blockade runners which were usually painted in shades of gray and black as camouflage.