In a video posted on social media by former Croatia global Vukojevic, Vida was shown shouting "Glory to Ukraine" which had angered Russian officials given the past between the two countries.
In the video, which was then posted to Facebook by Ukrainian journalist Andrey Shakhov, Vukojević can be heard saying, "This is a victory for Ukraine and Dynamo", before Vida added, "Glory to Ukraine", a slogan used by Ukrainian anti-Russian nationalists.
For Russians, the sadness was mixed with pride on Saturday evening after their team - the lowest ranked in the tournament - was halted in its tracks to World Cup glory by Croatia in a dramatic quarter-final match.
Russian Federation equalised in the 115th minute when Mario Fernandes headed in a free kick from just outside the area after a handball was called against Croatia's Josip Pivaric.
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Federation Internationale de Football Association have began probe into Croatia's celebrations, after they eliminated Russia from the 2018 World Cup on Saturday night. It wasn't immediately clear whether they intended the video to be made public, or if it was a personal message later uploaded to YouTube by someone else.
Besiktas centre-back Vida, 29, and his former Dynamo Kiev and global teammate Ognjen Vukojevic recorded a video message following Croatia's win in Sochi. Football is outside politics. Ukraine's government accuses Russia of arming pro-Russian separatists who control part of the east of the country, and of backing up the separatist groups with Russian military forces. Vida was also shown a yellow card for taking his shirt off after scoring the goal.
"I also believe that today, regardless of which team wins, we will see an excellent and fair play and will see how to overcome everything in our daily lives that could create obstacles", Grabar-Kitarovic said.
There were suggestions earlier in the tournament that Switzerland duo Granit Xhaka and Xherdan Shaqiri could face punishment for their controversial celebrations against Serbia, but the pair have so far escaped punishment for their actions.
Croatia play England in the semifinals in Moscow on Wednesday.