The Dens Park side travelled north with a depleted squad after a positive Covid-19 test in their camp, with five further players deemed to be close contacts and also ruled out.
McPake refused to use the decision as an excuse for his side's 2-1 defeat, but felt strongly that the game should have been called off for the safety of both sides.
On the pitch, Dundee took the lead thanks to a spectacular Leigh Griffiths free-kick, but Ryan Hedges restored parity just five minutes later.
Lewis Ferguson netted what would prove the winner with a powerful second-half drive.
McPake said: "I felt that it's a bad decision, and I hope there's no backlash in terms of players going back to their families.
"I felt for Jim Goodwin through the week, and Dundee United had to go to Ibrox with a much different side from normal.
"It wasn't about the game today. It's a complete and utter lack of care for the individuals who have to go home to their families. For me it's lacking class.
"You see the way the English FA do it, and maybe the SPFL should take note.
"In terms of effort, what they gave, I'm proud in that sense. The goals we could have done better with, but I can't fault the effort.
Aberdeen boss Stephen Glass said: "It would have been easy to take our eye off the ball. I'm sure Dundee wanted the game to go ahead, regardless of trying to get the game put off.
"I don't think the fact they were down to the bare bones really mattered too much as they had a really good team on the pitch.
"Any time you go a goal behind it's difficult. The conditions weren't great, but I think you saw the mental toughness of the group and the quality too.
"Any time we showed intent we were a real danger, and from a goal behind it's good to come back and win."