When I read the phrase "rainbow nails", I was thinking of something more like this, with little tiny rainbow stripes on EACH nail:
The "official" gay flag has exactly SIX colors: красный, оранжевый, жёлтый, зелёный, синий, фиолетовый. (Ironically, in the context of this thread, the colors розовый and голубой DO NOT APPEAR on the flag!) There are also "non-gay" rainbow flags with more than six colors (the "peace rainbow" flag sometimes has 7 or 8 stripes).
If the athlete hadn't posted her picture with the tag #pride, I would never have guessed that this was some sort of gay propaganda, since her nails appear to have only three colors, and not a "whole rainbow". So, evidently, more than two colors + the word "pride" = insult to Russian feelings.