UEFA said in a statement the 40-year-old, who was livid at a stoppage-time penalty awarded to the Spanish side during the second leg last month, had been sanctioned for a direct red card and infringing "general principles of conduct".
Buffon, a World Cup winner in 2006, left Juventus at the end of the season after 17 years at the club and has still decide whether to retire or continue his career elsewhere.
Juventus, beaten 3-0 in the first leg at home, remarkably managed to claw back the deficit at the Bernabeu and the match was poised to go to extra-time when Real were awarded a controversial penalty by English referee Michael Oliver.
Cristiano Ronaldo converted and the 3-1 defeat meant that Real, who went on to win the competition for the third year in a row, qualified 4-3 on aggregate.
A livid Buffon jostled and ranted at the referee and was sent off.
Afterwards, he said the referee had been out of his depth, should have been in the stands eating crisps and had a "garbage bin for a heart".
Although generally admired as a model professional, Buffon was criticised for the outburst with former England forward Gary Lineker describing his comments as inflammatory.
Buffon, who played 656 games for Juventus, kept 300 clean sheets and won nine Serie A and four Coppa Italia titles with the Turin side, later apologised for the outburst while still insisting the decision was wrong.
"I stepped over the line and I am extremely disappointed. If I had seen him (the referee) two days later, I would have embraced him and asked for forgiveness," he said.
The defeat ensured Buffon would leave Juve without a Champions League winners’ medal, the only major title to have eluded him in his career.
(Writing by Brian Homewood; Editing by Ed Osmond)
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