Here is the specific fare rule and it is not unique to United (other airlines have similar rule for their cheaper RT fares). It is still a "Saturday night" stay requirement, but it is measured from arrival, NOT from departure.
MIN STAY
TRAVEL FROM LAST STOPOVER MUST COMMENCE NO EARLIER THAN THE FIRST SUN AFTER ARRIVAL AT TURNAROUND.
Because most flights to Europe are overnight flights and you do not arrive until the next day, you have to depart on a Friday to meet the Saturday night stay requirement at the destination. There's an exception here with the daytime flights to LHR where you can depart on a Saturday and still arrive on a Saturday and thus meet the Saturday night stay requirement. Since most domestic flights in the US are daytime flights (other than red-eye's from the west coast), you can generally depart on a Saturday and still meet the Saturday night stay requirement.
Now I understand what happened on my last trip to EDI. I had to leave on Friday to get an acceptable rate but then I had a hard time getting refunded for my Saturday night Hotel even though prior to departure it was cleared by my department that a night hotel at $260 was cheaper than the additional $1000 United wanted for my flight. Anyways, lesson learned. Next time, I will stay 2 additional nights at the end of the trip as it's way easier justifying hotel stays during week-days.
Last edited by WineCountryUA; Sep 26, 2017 at 12:54 pm
Reason: merging consecutive posts by same member