Not according to this site.
http://www.angelfire.com/wv/wasec5/formations.html
Each Union or CSA infantry regiment was made up of 10 companies, each (theoretically) of 100 soldiers.
There was some difference in cavalry regiments, apparently. A Union regiment had 12 troops of 100 men each, while a CSA regiment had 10 troops.
Throughout the war, and on both sides, there were also units that varied from this ideal, even in theory. In practice, as stated, regiments very often were 50% to 80% under-strength at any given time. I believe CSA was more likely to assign new recruits to an existing regiment than was the Union, making their regiments larger, as you say, on average.
Here’s another link. It claims Union heavy artillery/infantry regiments also had 12 companies.
http://www.civilwarhome.com/armyorganization.htm
According to this site, while Union and CSA regiments tended to be about the same size (always in theory), formations above the regiment level tended to be larger in the CSA, presumably because CSA brigades and divisions had more regiments in each.
http://www.civil-war-journeys.org/military_terms.htm
As you can see, there’s more than a little difference of opinion out there on the subject.
This is a thread about cavalry which this discussion about infantry is probably not exactly applicable, as some cavalry were partisan fighters which who knows how many were in each "unit".