In a statement, however, Swedish officials stressed that they continue to maintain close links with the gay rights movement in Russia.
Some 20,000 people had supported a petition on Facebook calling for the embassy to fly the flag during Stockholm Pride.
Defending the embassy's decision, Swedish Foreign Ministry spokesperson Lena Tranberg told newspaper Svenska Dagbladet that "there is a praxis we have to follow, and that is that we fly the Swedish flag and none other."
The increasingly harsh climate in Russia surrounding gay rights is one of the topics that will be discussed at the Stockholm Pride festival. A new law, signed recently by Russian president Vladimir Putin, bans "homosexual propaganda."
On Monday, Swedish Foreign Minister Carl Bildt described the law as "repulsive and "inhuman" in a post on his Twitter feed.
Russia's embassy in Sweden later responded to Bildt's comments in a written statement, describing the accusations as false and claiming that the new law does not infringe on the rights of LGBT individuals.
The embassy's press attaché Alexander Pashedko referred to Russian society's "own set of moral values" and said that Russia does not "want to create conditions for reverse discrimination, where a small group of citizens are given the right aggressively to market their opinions to children which go against the views held by the dominating majority of Russians."