Around two dozen Dons supporters staged a protest outside Pittodrie following Saturday's Scottish Premier League win over Inverness.
Some fans are unhappy with last month's Scottish Cup defeat to Dunfermline, the second straight season Aberdeen have crashed out to lower-league opponents.
Calderwood said in the Evening Express: "It's not pleasant to know people feel that way about you, everyone would rather be praised.
"But I would rather speak about how well the majority of the fans have backed us over the last few weeks.
"The Aberdeen FC fans have been brilliant when you consider how badly we let them down against Dunfermline."
Gary McDonald, who netted the winner in Saturday's 1-0 victory to keep the Dons in fourth place, added: "You never like to see protests and things like that, so we wanted the win even more than normal for the manager.
"Thankfully, we got that to make sure we finish in the top six again this season.
"Now we aim to get into Europe and, to do that, everyone has to pull together."
Celtic 2/7, Draw 9/2, Aberdeen 9/1