"I deeply regret the words I used in a verbal altercation on Sunday,'' the 22-year-old McGinnity said in a statement.
"The heat of the moment is no excuse for over-stepping the mark and I sincerely apologise to the Melbourne player who my comments were directed at.''
Eagles chief executive Trevor Nisbett said the club moved as quickly as possible to resolve the matter with Melbourne.
"The comments made by Patrick do not reflect the values of our football club and we acted quickly when we became aware of the issue,'' Nisbett said.
"We have high standards as a club, which we have continued to enforce in recent years and we are disappointed Patrick stepped outside those guidelines.
"We will continue to educate our players on their responsibilities around this and other AFL policies.''
In June, Western Bulldogs forward Justin Sherman was suspended for four matches and fined $5000 after pleading guilty to racially vilifying Gold Coast youngster Joel Wilkinson.
- AAP