The former Dundee and Rangers player was a surprise choice for the post following the departure last week of Barry Smith who left the club 15 points behind St Mirren at the bottom of the Clydesdale Bank Premier League table.
Some supporters walked out of a fiery meeting with the Dundee board on Saturday after it was confirmed that the 51-year-old, whose only previous management experience was with Clyde and which ended in 2009, would be boss until the end of the season.
However, Brown said: "There's a lot of guys out there who have probably been critical about the appointment, who've probably put their CV in last week and are saying it's the wrong appointment, maybe because they didn't get the opportunity.
"There's a bitterness there. I don't know the reaction from fans because I don't listen to the radio or read the papers. I've been too long in the game but I've had pals phoning me and saying 'aye, you thought he was your pal....'
"I didn't want to say to them but he (one of his critics) was one of the first who put himself in for the job after singing the previous manager's praises. That's football. There's a lot of guys with a lot of coaching badges that are out of work. It's easy for them to have a pop but just wait and see what happens."
The former Hamilton player, however, admitted that he was "very surprised" to be offered the job by chief executive Scot Gardiner but believes keeping Dundee in the SPL would be the best achievement in his successful football career, in which he enjoyed huge success at Ibrox.
"I was out of football for a year and got a phone call on Wednesday night from Scot, asking if would I help Dundee out and that's what I'm here to do," he said. "To keep them up would be the best achievement ever."
Gardiner claimed that the supporters who were critical of the board's decision to bring in Brown "weren't representative of the whole support" but revealed he had apologised to his new manager for the stormy reaction to his return to the club where he was a fans favourite in the early 1980s.
"I think John's been caught in the crossfire a bit and I've apologised to him for that," he said. "On Saturday I think I said, 'sorry about this John'."
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