The Russian Olympic Committee has sent a letter to the head of the International Ski Federation (FIS) complaining of biased judging.
Koroleva held a big lead after the first jump of the final but slipped to fourth overall after the second jump.
Australia's Alisa Camplin won the event ahead of two Canadians, Veronica Brenner and Deidra Dionne.
Russian Olympic Committee spokesman Ghennady Shvets said: "We haven't lodged an official protest.
"We simply sent a letter to FIS in the hope of shedding some light into the judging process, which we consider unfair."
"We feel she was judged fairly and this is a non-issue"
FIS spokesman Chris Robinson
FIS freestyle committee chairman Chris Robinson said its rules do not allow for scores to be protested, and the FIS "felt the judging was very good."
Koroleva got the second-best raw score of the day on her second jump.
But the degree of difficulty on the jump was much lower, which caused her to fall to fourth.
Shvets said the Russians would not file an official protest since there was no chance of her being awarded a medal.
"We're complaining because we think there's a bias against Russian athletes," said Shvets.
"It cost us medals in the past, and it cost us a medal now. That's not acceptable."
Russia was involved in the 'Skategate' judging scandal earlier in the Olympics.
French figure skating judge Marie-Reine Le Gougne was suspended after she voted for the Russian team of Elena Berezhnaya and Anton Sikharulidze in the pairs competition.
The Russians beat Canada's Jamie Sale and David Pelletier for the gold by a 5-4 split.
But Le Cougne later admitted she had been pressured to vote for the Russians, and the IOC awarded a second gold medal to the Canadian pair.
In a separate development, Lithuania have protested about judging after their figure skating team were placed fifth in the free dance competition.