Spectators at Wednesday's match in Moscow booed and whistled the Croatia centre back, after a video posted online earlier in the week showed Vida, a former player with Ukrainian club side Dynamo Kiev, saying "Glory to Ukraine!".
"I know I made a mistake and I would like to apologise again to Russian people," Vida said, speaking in Russian in a live post-match interview with the Rossiya 24 state TV channel late on Wednesday.
"I'm sorry. That's life. You have to learn from your mistakes," he added after their 2-1 extra-time win.
Russia's relations with Ukraine have been fraught since Moscow annexed Crimea in 2014 and backed pro-Russian separatists in eastern Ukraine.
Russian Foreign Ministry spokeswoman Maria Zakharova on Thursday condemned the video as "not clever".
"There is soccer, and there is politics, and the two are different things," Zakharova told reporters, adding that the ministry had seen Vida's apology for the incident.
The booing and whistles rang out each time Vida touched the ball in the semi-final against England, starting about 30 minutes into the first half.
It did not appear to be coming from sections of the stadium where England supporters were concentrated. There were significant numbers of Russians in the stadium.
World soccer's governing body FIFA ruled that Vida's comments in the video were not in keeping with tournament rules about political neutrality and its disciplinary committee issued a formal warning to the defender.
FIFA hit Ognjen Vukojevic, a member of Croatia's coaching staff who appeared in the video with Vida, with a 15,000-Swiss-franc (11,400 pounds) fine. Vukojevic was fired from his role with the national team.
In an earlier statement released by the Croatian Football Federation, Vida, 29, said his comments were not intended to be political and were related to his former career with Dynamo Kiev.
"Our intention was not to offend anyone... I sincerely hope that this message will not be understood as anything else but an expression of gratitude to our friends in Ukraine for their support," Vida said.
Croatia will face France in the World Cup final on Sunday.
(Reporting by Maria Tsvetkova and Christian Lowe; Writing by Polina Ivanova/Andrew Osborn; Editing by Toby Davis)
