Hamilton published confidential telemetry of his qualifying lap at Spa to show a speed comparison to Button, whose low downforce configuration was better suited to the track.
However, the image of the data provided other valuable insights in to the performance of the McLaren car, such as ride height and speed data.
One rival team boss told AUTOSPORT that it had been 'gold dust' for his engineers in understanding the performance of the MP4-27.
McLaren team principal Martin Whitmarsh said the outfit asked Hamilton to delete the image, but that it does not intend to take the matter any further.
"I think he made an error of judgement, and we asked him to take that one down and he did," explained Whitmarsh. "I think he realises that, and I don't think it's had an impact on his weekend."
When asked if he felt it was a breach of confidentiality, Whitmarsh said: "I don't think... no.
"It would be interesting to see how other team principals would deal with it. But I think no, it was an error of judgement, it was corrected fairly quickly and apologised for. We move on."
McLaren technical director Paddy Lowe doubted that rival teams would have gained much from the data, but says it was still wrong for the information to have been posted.
"Like all of these things the internet has an infinite memory and I am sure the tweet has been taken down," he said.
"I have had several engineers approach me and it wasn't great, but in reality it is more about that we don't share data with the world. The actual data in there isn't going to be any great use to anyone, I don't think there is much damage done.
"The actual mistake that Lewis made, which he understands, is that he didn't really appreciate the nature of that information.
"The engineers don't like to see that because we spend our lives trying to keep things like that secret. So it is more what it represents."
Red Bull team principal Christian Horner said if one of his drivers had acted in such a manner than it would have been treated much more seriously.
When asked how he would have felt if one of his drivers had done it, he said: "They would not have been able to. It would be a breach of confidentiality effectively."