Vida's video after the game on Saturday appeared online and led to criticism from Russian politicians and a warning from FIFA's disciplinary committee.
Relations between Ukraine and World Cup hosts Russia remain fraught after Russia's 2014 annexation of the Crimean peninsula and its backing for a pro-Russian uprising in the east of the country.
The 29-year-old Vida previously played for Ukrainian club Dynamo Kiev along with Croatian coaching staff member Ognjen Vukojevic.
"I regret that some media representatives have interpreted our communication in such a manner. It was definitely no political message, but a simple thank you for all the support from Ukraine, where Vukojevic and I spent a number of years," Vida said in a statement released by the Croatian Football Federation (HNS).
"Our intention was not to offend anyone. Throughout my career, I have had team mates from many countries and I respect them all, and just as I have many friends in Ukraine, I have a number of them in Russia - and I am proud of all of them.
"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 - not in the match against Russia, but during the entire World Cup," Vida added.
The HNS said it had asked Vida and Vukojevic and all the Croatia players "to refrain from any messages that could be politically interpreted in the future".
Croatia face England in the semi-finals in Moscow on Wednesday.
(Reporting by Simon Evans, editing by Ed Osmond)
