Most democratic countries use civil law, actually, though I don't think that's particularly relevant. There's a much stronger correlation between language and system of law than there is between form of government and system of law. Note the similarity between [0] and [1], and the complete lack thereof between [1] and [2].
What common law countries call a "civil case" is not the same was what civil law countries call "a case". It's confusing.
As for the distribution of languages and laws, it's largely due to history. Civil/code jurisdictions show the extent of the former Roman empire and later the influence of Napoleonic Code. Common law countries are almost universally within the Anglosphere and have a common legal heritage commencing in the 1066 Norman Conquest.
[0] https://en.wikipedia.org/wiki/File:Anglospeak%28800px%29.png
[1] https://en.wikipedia.org/wiki/File:LegalSystemsOfTheWorldMap...
[2] https://en.wikipedia.org/wiki/File:Democracy_Index_2011.png