Re: which one is correct? - release day or release date
Welcome to Using English.
In general, it's easier to answer questions like this when we have a full sentence.
However, "release date" is very standard.
I"m sure it's possible to write a sentence where "release day" is the better choice, but such a sentence is not occurring to me at the moment.
I'm not a teacher, but I write for a living. Please don't ask me about 2nd conditionals, but I'm a safe bet for what reads well in (American) English.