Scottish Rugby issued a statement Thursday saying it was "appalled" by the abuse suffered by Jones.
"The disgusting behaviour of those involved does not represent the values of our sport or its fans," the statement said.
"The dignity Eddie and the England team showed on Saturday is in stark contrast to this ugly incident."
Jones had suggested on Wednesday that everyone connected with rugby needed to be careful about the language they used.
Prior to the Calcutta Cup clash, Scotland great Gavin Hastings, a former British and Irish Lions captain, gave an interview in which he said: "As a supporter of one of his opponents you just want to rub his face in the dirt."
Meanwhile current Scotland prop Simon Berghan said "everyone hates England".
"It magnifies that if you're in a position of responsibility you've got to be careful what you say," said Jones.
"Because if you talk about hate and you talk about rubbing peoples' nose in the dirt, and all those sorts of things, it incites certain behaviours and are they the sorts of behaviours that we want to see?"
Jones added he had enjoyed meeting fans but that this might change in the light of his recent experience.
"I never knock back a request for a selfie unless I'm racing to somewhere," he explained.
"So I try and do the right thing by the fans, but if this happens then you've got to have a look at your own safety."