Football shows were pulled at the last minute when presenters and commentators walked out in support of Gary Lineker.
The Match of the Day host was suspended on Friday after he criticised the government's controversial asylum policy.
Mr Davie denied he was pressured into the move by the government.
The BBC boss admitted it had been a "difficult day" for the corporation but said "we are working very hard to resolve the situation".
Interviewed by BBC News, Mr Davie said "success for me is getting Gary back on air", adding he was prepared to review impartiality rules for freelance staff like Lineker.
The director general said he would "absolutely not" be resigning but admitted "this has been a tough time for the BBC".