The incident occurred when the official called a foul for a challenge by forward Kylian Mbappe on defender Eric Garcia, denying Real a final chance to find an equaliser moments after Barcelona's Jules Kounde scored the winner in the 116th minute.
Ricardo de Burgos Bengoetxea's decision sparked an outburst of rage among the Real Madrid players, especially Rudiger, who was on the bench after being substituted earlier.
The German saw a straight red for his reaction and had to be restrained by his teammates and coaching staff to prevent him from bursting onto the pitch and throwing bags of ice at the official.
"There's definitely no excuse for my behaviour last night. I'm very sorry for that," Rudiger said on his social media accounts on Sunday.
"We played a very good game from the second half on. After 111 minutes, I was not able to help my team anymore, and before the final whistle, I did a mistake.
"Sorry again to the referee and to everyone I have disappointed last night."
His teammates Lucas Vazquez and Jude Bellingham, after the final whistle, were also shown red cards for unsportsmanlike conduct, although the centre back is the one most at risk of facing a severe sanction for his actions.
Rudiger could be suspended for a range of four to 12 matches according to the Spanish FA regulations, and, being considered a serious sanction, he would have to serve the ban in all domestic competitions, including LaLiga.
Real Madrid, last year's Spanish and European champions, have the league remaining as their only chance for silverware this season. They are four points behind leaders Barcelona in the standings with five fixtures remaining.