Inter have lost their battle to change the dates and will now face Juve next Thursday with Milan entertaining Udinese in their last-eight tie 24 hours earlier.
The row has overshadowed the build-up to Sunday's Serie A derby between leaders Inter and second-placed Milan, who can cut the gap to three points with a game in hand if they win.
"Given several Italian teams are competing in three competitions, the calendar cannot undergo any more changes in this phase because there are no more windows from here to the end of the season," the Italian league, which also organises the Cup, said in a statement on Wednesday.
"As for the state of the San Siro pitch, where next week two games will be played in two days, the league will make sure the issue is given the utmost attention."
Milan moved the Udinese game to next Wednesday from this Wednesday so they would have a free week like Inter before the derby.
Inter objected to the switch, saying the turf at their shared San Siro stadium would be too cut up for a second match in two days, so Milan proposed the drawing of lots to see which of the city's clubs played first.
Inter rejected that idea and investigated whether their quarter-final could be put back, only to stick to a Thursday kickoff because of television commitments.
The argument has added an extra dimension to the usual heated rivalry ahead of a Milan derby.
Inter coach Jose Mourinho had joked that if Milan could put off a game so easily, the Italian league should also postpone the derby given his side's injury problems.
Midfielder Sulley Muntari was the latest Inter player to pull up in training on Wednesday.
