Constitution Day
Possibly because Denmark has been an independent country since times immemorial, the country doesn’t have a national day, so Constitution Day is the closest you get. It’s not a day of big celebrations, though — in general, people gather to listen to politicians speaking about the value of democracy, but that’s about it.
However, I do like the fact that the country has chosen to celebrate its constitution, rather than some battle or royal event. I guess the Scottish Constitution will come into force on the day Scotland becomes independent (March 2016?), so constitution and independence might become interwoven for Scotland, too.