It was identified as one of the most romantic scenes ever on Gilmore Girls, at least by the readership of the GilmoreGirls.org forum. It’s at the end of Episode 516, “So… Good Talk.”
Lorelai is sitting alone on her couch, watching A Star is Born, missing Luke, a couple weeks after he told her, “I can’t be in this relationship. It’s too much.”
On the TV, Judy Garland as Esther Blodgett is singing:
The night is bitter.
The stars have lost their glitter.
The winds grow colder.
Suddenly you’re older.
And all because of the man that got away.
A knock at the door. Lorelai opens it. Luke is there, but also not Luke. He needs no time to think, no time to process. He does not dance around why he’s there. He doesn’t struggle. There’s no stammering, no silly excuses, no polite chit-chat. With strong arms he pulls her to himself, brings her lips to his, and fills her soul with life again.
According to Wikipedia, Harold Arlen and Ira Gershwin wrote the song “The Man that Got Away” especially for this film. It is one of the all-time best performances is one of the all-time best musicals in film history.
Ironically, the Judy Garland film A Star is Born has a history. It was Judy Garland’s big comeback and the last film she ever made for Warner Brothers. It was a hit, then butchered, then restored. TheJudyRoom.com tells this interesting story.
I myself have not yet seen A Star is Born, believe it or not. But like Lorelai, I’ve Netflixed all three versions of the film. I should get them this week, and I’ll write more about them then. Stay tuned.
-TimK
Ooh! Yes, April, that is definitely in competition for the most romantic moment. The song they were waltzing to is “Reflecting Light” by Sam Phillips.
But my favorite romantic moment of that episode was when Luke asks Lorelai out at the end, and Lorelai says yes, and then she turns around with an expression on her face that screams, “Did he just ask me out on a date?!”
Part of my problem is that there are so many great moments, I have trouble choosing my favorite. However, my favorite romantic Gilmore Girls moment is probably still the “Thousand Yellow Daisies” at the end of the first season.
-TimK
I agree that both “The man that got away”, and the song lorelai and Luke dance to are two of the most romantic moments from Gilmore girls. I just watched the episode with “The man that got away” the other day, and instantly had to download it (not to mention im a big Judy Garland fan as well.) 
competition for most romantic moment
when luke and lorelai danced together at liz and t.j.’s wedding, the song that luke and lorelai danced to is in strict competition for the most romantic moment…anyone agree?