1) The longest day in St.Petersburg is the 22nd of June, which is the day of the Summer Solstice
2) The longest day in St.Petersburg is on the 22nd of June, which is the day of the Summer Solstice
Do we need "on" in the sentence above before the date?
No. In this case it is incorrect to use "on." If you were saying "The Rolling Stones concert is on..." then you would be talking about something happening on that date. Then the "on" would be optional.
In this case, you are not talking about something that happens on that day, but are talking about the day itself.