There's nothing wrong with the word "Gay" so it is perfectly to okay to ask "Are there Gay Days on the ship?" The answer is that there have not yet been any events similar to those at WDW on
DCL. That said, DCL is open to everyone and you definitely see all kinds of families. If you will be offended by a same-sex couple holding hands or giving each other a quick kiss be warned that you may see that on board...just as you'd see the same with heterosexual partners. I have never noticed any obnoxious or inappropriate displays of affection from any couples whether they are same-sex or opposite-sex, although I hear that some teens have been found in compromising positions from time to time.
The bottom line is that gay people do sail on DCL and there will probably be some on your sailing, although you may never know it. The same is true of all cruise lines and any other place you may go. There are not, however, any sponsored events on DCL that I am aware of.