Ashes-winning skipper Vaughan, now a commentator with BBC Radio's Test Match Special, made his stinging comments as England lost seven first-innings wickets for 62 runs to be bowled out in little more than 50 overs for 205 in reply to the Proteas' 335.
Root's 78 was the only fifty in a meagre total that left England well behind in the game.
"The England batting has been appalling," said Vaughan, who played alongside Root's father Matt at the Sheffield Collegiate Club in Yorkshire and has been a friend of the family since Joe was a boy.
"Maybe it's a lack of respect about what the game is," said Vaughan, one of the leading Test batsmen of his era.
"They look like they are playing a Twenty20 game. They have this approach of attack, attack, attack. There is no thought or feeling of seeing off a bowler or wearing a team down."
England's second innings was even worse, with Root's men bowled out for just 133 in pursuit of a world-record target of 474 with more than a day to spare on Monday as South Africa levelled the four-match series at 1-1.
This was England's sixth defeat in their last eight Tests and Root, in charge for just the most recent two, was taken aback by mentor Vaughan's comments.
"I think that's very unfair -- I can't believe he's actually said that.
"We pride ourselves on winning series like this, and unfortunately we've played poorly this week," Root added.
But the Yorkshireman accepted his side ought to be able to bat in different styles according to the match situation, whatever the individual preferences of particular players.
"We need to make sure we learn quickly ... (so that) if we are in a similar situation (in the third Test) at The Oval, for example, we play it slightly smarter," said Root.
"Part of Test cricket is trying to find a balance of both (attack and defence) -- and even if it's not your strength, you build it into your natural game," he added.
"You have periods where you can absorb a bit of pressure and find the right moment to apply it when chances come along.
"We are capable of doing it and have done it previously, but this wasn't a very good example of it."
Meanwhile the 26-year-old Root, who led England to a 211 -run series opening win at Lord's in his first Test as skipper, said this huge loss had been a chastening experience.
"I definitely feel older -- it's been a tough week," he conceded.
"After that, we couldn't find a way to get back into the game."
The star batsman added: "We're a side that doesn't like giving anything away, and unfortunately today that wasn't a fair representation of how good we are as a team."
By contrast, just about everything went right for Root's opposing captain Faf du Plessis, who did not play at Lord's following the birth of his first child.
"Pretty much from the toss through to the last ball we bowled, we were in control of the Test match," said du Plessis.
England number three batsman Gary Ballance, out for just four on Monday, was due to have an X-ray on his left index finger after being struck by a Morne Morkel short ball.
(This story has not been edited by Business Standard staff and is auto-generated from a syndicated feed.)