Vaughan, the former England captain who has moved into the media, suggested in the run-up to the Test that England should consider dropping Broad (or, to a lesser extent, James Anderson) in order to "ruffle some feathers".
But Broad hit back after his good performance on the first day of the Leeds Test, saying that Vaughan "doesn't know what the changing room is like" and didn't have much insight.
"I think it was a bit of a wild guess," Broad said, before suggesting Vaughan was motivated by self-publicity. "It's [about] personal columns and radio shows that need 'likes' and airtime, isn't it?"
Broad also confirmed that he had phoned Vaughan to register his "disappointment" at the comments. But Vaughan, responding on BBC's Test Match Special programme, defended his views and suggested Broad had spoken prematurely.
"You have to be careful when you choose a time to come out and attack like Stuart did last night," Vaughan said. "England are still 72 runs behind Pakistan. They haven't won this Test match yet.
"The comments last night were geared as though they had won the game. He's a senior member of the team and I don't think it was the right time to plan that attack.
"With the way the England team have played for a long period of time, they have had a great first day. To come out speaking like that after one good day after having only four or five good days in their past eight or nine Tests.
"You have to be careful in sport that comments can come back to bite you but, on the other hand, it is entertaining. And that is what we want in sport. My only concern is that they are speaking like that after one good day.
"I got the sense last night it was 'you can't criticise me. I'm Stuart Broad and I've been in the team for a long time.'"
Nick Compton, who came in for a lot of criticism from Vaughan during his spell as an England player, also spoke out on Twitter saying: "About time someone shut that t&@t up." He later removed the Tweet.
"The reasons why I said they should consider [dropping Broad] is that I felt the England Test team needed to ruffle a feather or two," Vaughan continued. "It has been very comfortable for a long period of time and one of the options might have been to break up the senior bowling pair.
"If the criticism over the last few days has geed them up to go out there and prove us wrong, great.
"I think the Test team for too long has been a lovely place. Lovely and comfortable. They win games and series when the ball swings around, but they don't do it on a consistent basis.
"So, you have to look at every aspect of the team to think about how you can trigger them into a more consistent performance."