Springboks director of rugby Erasmus made the claim on social media.
Vunipola tried to help Cheslin Kolbe off the floor after the South Africa wing had been tackled by Ali Price.
"He seemed to be all right. He played on in the game - so it wasn't that reckless, was it?" said Vunipola.
Erasmus posted the clip on his Twitter account, where he criticised the action of the England prop who had come on as a second-half replacement.
He tweeted: "Cheslin is obviously played in the air and clearly not direct into touch. More importantly for youngsters watching this clip, please never move or touch an injured player on the ground - it's reckless and dangerous."
Vunipola said he had been bemused by the backlash over the incident but added that he had been targeted in the media on a previous Lions tour.
"The same happened in 2013 when someone - Bob Dwyer I think it was - said something about my scrummaging," said the 30-year-old. "Maybe this is the first time a coach has said something like that on social media.
"I remember trying to get the ball in but I saw that he'd gone down. I guess if he was really hurt then it was a bit reckless, but I felt like the collision wasn't that bad.
"I saw that people were putting it up [on social media] and mentioning it. We were behind at the time and wanted to get some tempo in the game, so I wanted to get the ball off him.
'If I did hurt him, I do apologise - but in the heat of the moment, you just react as you would naturally."