Celtic, who avenged their defeat to the champions in a last-16 match last season, now face the prospect of UEFA, football's governing body, imposing Draconian sanctions, including the deduction of points or expulsion from the competition.
Tensions were running high at Celtic Park as the hosts battled for their first victory in the group stage of the competition as the match entered the 90th minute.
Celtic's Scott McDonald had just scored to make it 2-1 when the fan ran triumphantly from the Celtic end of the ground across Dida's path in the goalmouth before making contact with him.
There are varying accounts of what happened next - some commentators say Dida was "caught around the throat" by the fan, others say he was "stroked across the chest".
Dida, 33, took two or three steps in pursuit of the fan and then collapsed on the ground clutching his face. The club's medical staff rushed to attend to the Brazilian and he was stretchered away.
Moments before the incident, Celtic players had gathered in a corner to celebrate their goal when the fan seemed to make an attempt to join them.
Strathclyde Police said they were investigating, but no-one had been detained last night.
The game continued, but Dida was replaced by a substitute. "I don't think it's serious," said the AC Milan coach, Carlo Ancelotti.
Two seasons ago, Dida was hit by a flare during a Champions League quarter-final between AC Milan and Inter Milan.
That game was abandoned and UEFA awarded a 5-0 aggregate victory to AC Milan, who reached the semi-finals. Inter Milan were fined 132,000 and had to play four European games behind closed doors.
Gordon Strachan, the Celtic manager, refused to discuss the incident. Celtic are to hold their own investigation.
UEFA's director of communications, William Gaillard, said the body would wait for reports from the referee and match delegate before acting.
Peter Rafferty, president of the Celtic Supporters' Affiliation, said: "We are disappointed to have our elation at the result turned into something like this."