The goal gave Bayern a crucial foothold in the tie, and the German side need just a 1-0 win at home in a fortnight to go through.
The former Barcelona player said his reaction was sparked by previous run-ins with some Real Madrid players.
"I want to apologise to the Real supporters, I have got nothing against them, only against some of the players," he said.
The 29-year-old left Barca for Bayern at the end of last season after helping the Catalans win both the Spanish title and the Champions League.
Real Madrid captain Raul and Ruud van Nistelrooy said Van Bommel had apologised to them for the gestures but Iker Casillas said the Dutchman should have shown more control on the pitch.
"You can do whatever you want in the dressing room, but you have to know how to behave on the pitch," the Real keeper said.