The fiery Argentine was caught on camera repeatedly provoking the Brazilian winger throughout the match, reportedly telling him, "Florentino will kick you out - remember what I'm telling you," in reference to speculation about Vinicius' future and contract discussions as his current deal approaches expiry.
"I would like to apologise to Mr Florentino and Mr Vinícius for the incident they witnessed," Simeone told a press conference on Monday, ahead of his team's Copa del Rey round of 16 tie with Deportivo La Coruna.
"It was wrong of me to put myself in that position, and I accept that it was wrong. The team that wins deserves to go through; they deserved to go through."
Simeone was pressed on whether his behaviour had crossed a line.
"I apologise, but I do not ask for forgiveness. I have nothing more to add," he said.
The situation escalated during Vinicius' second-half substitution, as boos rained down from sections of the crowd. As the struggling winger - enduring a 19-game goal drought - walked off, a laughing Simeone gestured toward the crowd, saying, "Listen, listen!"
Real Madrid won 2-1, securing their place in the final, which they lost 3-2 to arch-rivals Barcelona. Vinicius scored a stunning goal in the first half of the final.
"It was not an example of a good sportsman," Real Madrid manager Xabi Alonso said of Simeone's behaviour.
Atletico are fourth in the LaLiga standings, 11 points behind leaders Barcelona.