Fight Face - S6-E2
Other mistake: While in Luke's diner, Luke hands Lorelai a cup of coffee in a 'to go' cup. When Luke puts the coffee in front of Lorelai, the cup makes a hollow sound when it hits the counter, obviously indicating that there's no actual coffee in the cup.
Chosen answer: Lorelai and Christopher get married impulsively in Paris but they get divorced shortly after returning home. I don't know the exact number of days or weeks but they definitely weren't married for more than a couple of months.