Dungy's decision to attend the group's banquet to receive the award and give a speech has upset gay-rights groups, The Indianapolis Star reports.
"When you start helping groups like this raise money, it becomes problematic to me," said Cyd Zeigler, one of the founders of OutSports.com. "Be very clear about this: Tony Dungy is raising money for this organization. He is the face of the event."
The team released a statement saying that Dungy is free to speak to any group that he wants. His decision does not imply that the Colts support any of the institute's goals, the statement said.
The institute is affiliated with Focus on the Family, the national group founded by James Dobson. Dungy accepted the invitation in December.