Zimbabwean McGregor suffered a fatal heart attack on the ninth hole at Santo da Serra and although an indefinite suspension of play was initially announced, the decision was later taken to resume the event, which had already been reduced to 36 holes due to bad weather.
Many players and caddies criticised that decision and O'Grady held an "emotional" meeting with the European Tour Caddies Association during the Spanish Open this week, a meeting also attended by players' tournament committee chairman Thomas Bjorn.
In a statement on the European Tour website, O'Grady said: "We had a full and frank meeting with chairman Gerry Byrne and his committee, a meeting which was understandably emotional at times and one during which I apologised to them for the hurt and upset caused by events in Madeira.
"I completely understand the views of people who say that we should not have carried on, but it was a terrible situation for anyone to be in and the decision to finish the tournament was not taken lightly, either by myself or by the tournament officials on the ground in Madeira.
"However, that decision is in the past and the important thing now is that we continue to work with Mac's family and friends - as we have done from last Sunday - to assist with arrangements surrounding the funeral, which will take place in Madeira next Thursday, the same day as we wear 'Black for Mac' at Wentworth (during the first round of the BMW PGA Championship).
"I have also personally instructed a review of how we deal, operationally, at tournaments with situations such as this so that we can ensure the lessons of Madeira are learned."
O'Grady also paid tribute to Alastair Forsyth, whose bag McGregor was carrying at the time.
"Throughout this whole process Alastair has behaved like a true gentleman in incredibly challenging circumstances," O'Grady added. "He accepted our decision to play on in Madeira with grace and dignity and he should command enormous respect for the way he has conducted himself throughout this difficult week."
Gerry Byrne, chairman of the European Tour Caddies Association, said: "We as a committee have faced an extremely tough week. Dealing with the passing of our colleague and friend during a tournament is particularly upsetting.
"While we understand that decisions have to be made at very short notice, it will come as no surprise to anyone that all European Tour caddies felt the wrong one was made in Madeira.
"We went into the meeting unhappy and it was emotional at times, but George's humility and honesty in dealing with tough questions was greatly appreciated and it reassured us of the caddies' important position within the Tour.
"We can now look forward to celebrating Iain's life next Thursday and strengthening our links with the European Tour moving forwards."