Broad claimed that Vaughan has little insight into the England dressing room after the former England captain said that he should be dropped following the defeat to Pakistan in the first Test at Lord’s.
Broad said there “wasn’t a huge amount of logic” in Vaughan’s comments and revealed that he phoned him to express his disappointment.
But Vaughan hit back at Broad’s response, made on the opening day of the Headingley Test, and suggested that the player believes he is untouchable.
“I got the sense it was, ‘You can’t criticise me, I’m Stuart Broad, and I’ve been in the team for a long time’,” said Vaughan. “You have to be careful in sport that comments can come back to bite you.
“The reason why I said they should consider dropping Broad is that I felt the England Test team needed to ruffle a feather or two.
“Look at Sir Alex Ferguson, the greatest sporting leader and man-manger of our time; he ruffled feathers to try to regenerate and ignite teams.”
While the Vaughan/Broad row rumbled off-the-field, with Vaughan making his latest comments as a crowd of 13,084 waited patiently for play on Saturday’s day two due to steady rain, on it England strengthened their hand when the action finally began at 2.45pm.
Resuming on 106-2 in reply to 174, the hosts advanced to 302-7 in the 59 overs possible, nightwatchman Dominic Bess top-scoring with 49 and Jos Buttler finishing unbeaten on 34.
It was not quite the dominant batting display that England were hoping for, not helped by conditions that still assisted the bowlers, but it built on their efforts of the first day, when Broad, James Anderson and Chris Woakes each took three wickets.
When England called-up Bess for the opening Test, they were under the impression that they had picked a promising spinner.
That might still prove the case, but there is clearly more to the Somerset man’s armoury than his ability to rip an off-break.
At Lord’s, Bess scored a fighting 57 as England flopped to an innings defeat. At Headingley on Saturday, he was no less effective, looking as comfortable and composed as any of the established top-order batsmen.
In cloudy and humid conditions with the floodlights on, Bess quickly showed his skill, cannily uppercutting Hasan Ali for four before imperiously on-driving Mohammad Abbas to the boundary.
He lost his captain, Joe Root, with the score on 138, the Yorkshireman edging to the wicketkeeper off Mohammad Amir, having struck 45 from 72 balls with six fours after beginning the day on 29.
Bess and Dawid Malan added 62 in 18 overs before Malan fell to the seventh delivery after tea, edging a ball of extra bounce to first slip off Amir.
England fell to 212-5 soon afterwards when Bess was finally undone by a ball from leg-spinner Shadab Khan that turned and bounced sharply, edging into the hands of the solitary slip.
It should have been 220-5 but Jos Buttler was badly dropped on four at mid-wicket by Hasan off Khan, the ball continuing to the West Stand boundary to rub salt into wounds.
Buttler added 48 with Jonny Bairstow before Bairstow was sixth out, caught behind off Faheem Ashraf to the final delivery before the second new ball became available.
Chris Woakes made a jaunty 17 before being caught behind off the new cherry, Abbas getting one to nip off the seam and take the edge, wicketkeeper Sarfraz Ahmed accepting his fifth catch of the innings.
At stumps, debutant Sam Curran had 16 and the lead stood at 128.