Bale kicked out at Jonathan Viera twice before finally pushing the Las Palmas midfielder to the ground early in the second-half with the scores level at 1-1.
"I don't believe that it was deserving of a red card," the Welshman told Real Madrid TV. "I was shown the yellow for the foul, I was then pushed and I just pushed him back.
"I want to apologise to the team and the fans. I hope to learn from my mistake".
However, Cristiano Ronaldo's double salvaged a draw that keeps Real just a point adrift of league leaders Barcelona with a game in hand.
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"We can't look for excuses," said Madrid boss Zinedine Zidane. "What we can do is improve our performance. We are not happy with how we are playing.
"There are moments in the season like this. The past three or four games things haven't gone as we would want, but we are not going to look for excuses in the referees."
"I am not one to give him a clip round the ear, that is what I do to my children," said the Brazilian. "Bale is old enough to know if he has made a mistake or not.
"These things happen in football, but hopefully it doesn't happen again between now and the end of the season."
In a recurring theme of Madrid's season, a stunning late effort undid a lot of the damage caused by a below-par display.
The European champions trailed Villarreal 2-0 on Sunday before a rousing fightback in the final 25 minutes sealed a 3-2 win.
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However, having also lost 2-1 at Valencia last week after conceding twice in the first 10 minutes, Zidane insisted Los Blancos can't always leave themselves with a mountain to climb.
"We can't go over the top. There are things we have done well this year, but we have to analyse it and we will change things," added the Frenchman.
"We have shown we have character and we can do more. We have to improve particularly in our first-half performances. We have to think about the positives. We will analyse the negatives, but to come back from 1-3 to 3-3 is a huge effort."
A 6-1 thrashing of Sporting Gijon earlier on Wednesday saw Barca leapfrog Madrid into top spot and the two sides are still to meet at Real's Santiago Bernabeu in April.