Since having an "Independence Day" would somewhat imply that we had once been conquered and therefore someone to be independent from, Sweden doesn't have one. We also don't have a holiday that celebrates the founding or something similar, because Sweden, like God, has always existed - un-created and eternal.
Midsummer has been the de-facto national holiday. With its roots in ancient paganism it is second only to Christmas and tied with New Year's. The closest thing to an official day has been "The Day of the Swedish Flag" on the 6th of June, but that one is from 1916 (although with roots going back to the 19th century), was only made a national day in 1983, and wasn't a holiday. Until now, that is. In 2005 it was made an official holiday, and since celebrating a flag in isolation from the country it represents makes little sense it is called "Nationaldagen" or "The National Day".
I don't know if that means we've become more or less nationalistic, having given up our divine and eternal origins for a bog-standard national concept, but fact remains: We now have a National Day and it is on the 6th of June.