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.
Brown, though, was more concerned by the criticism his appointment elicited in some sections of the media and by some whom he believes would have been in for the job.
He 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."
Brown, who also had a spell as youth coach at Rangers, shrugged off the fans' negativity
"Barry is a legend at this club and what he did through administration and the 25-point deduction (thereafter) was remarkable but things happen," he said.
"I was 18 years at Rangers and when Paul Le Guen came in, I was out the door with no reason other than it was a new manager and it was time to move on.
"You are a number and that is it. But I am in here to do the best I can for Dundee."
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.
"It's a massive gap and people say there's a mountain to climb but people climb mountains every day.
"I want Dundee to be in the Premier League next year and that's what I'm putting over to the players.
"Nothing's up until it's mathematically impossible. There's a 15 point gap and 33 points to play for."
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'"
The Dens Park chief then refuted speculation that three members of the board were set to quit.
"There is no resignations," he said. "None. Zero. I am not expecting resignations.
"I don't think there's been a total breakdown between the board and the fans.
"There is no issue there. It is a fan-owned club and the board have taken a decision. That's perfectly normal.
"Saturday was very difficult for individuals, for five guys who don't get paid, to get that amount of abuse.
"After that a whole lot of rumours went about because people were probably shocked and stunned and things might have been said thereafter.
"But I spoke to every member of the board today and we are 100 per cent behind what we have to do and we believe John is the right man for right now."