Gibb was named in an investigation by Channel 4 that alleged Leave.EU pressured him and BBC editorial staff to drop an investigation into how it used Facebook targeting.
Emails leaked to Channel 4 claimed that in 2016 the group's co-founder Arron Banks and head of communications Andy Wigmore sent a "barrage of emails" to get the BBC story dropped, including an appeal to the then head of BBC Westminster.
Banks allegedly described the story as a "smear campaign" and threatened to make a formal complaint. In his book about the Brexit referendum, he claimed: "Robbie Gibb is being quite helpful and says he’s trying to hose it down".
Gibb, who left the BBC to become Theresa May’s director of communications in summer 2017, slammed the allegations, saying: "These allegations are ridiculous and without merit. I remain proud of my contribution to the BBC’s impartial coverage of the 2016 EU referendum campaign."
Wigmore appeared to confirm the authenticity of the emails on Twitter and threatened to release the personal phone number of the journalist involved on social media.
He said: "Dealing with stolen and hacked emails Channel 4, you’ve spaffed up against the wall any journalistic integrity you’ve ever had. Given Theresa May’s woeful handling of Brexit, I’m sure the 58,000 people that contacted Robbie Gibb by email phone and text are keen to get hold of him again, failing that we can always give them your number Jon (Snow)."
BBC - 'Gibb had no editorial influence'
The BBC dismissed the insinuation that its journalists and editors behaved improperly or succumbed to political pressure. It said Gibb was not in any way involved in the story, and did not have any say in the decision to investigate the issue or spike it.
The broadcaster maintains it killed the story because it was unable to establish that Leave.EU had targeted these organisations deliberately.
A spokeswoman added: "Due to this lack of evidence, the reporter involved did not run this story. This was a routine editorial decision made by him and his news desk. He would not allow a story to be spiked for any other reason, and it is a slur on his professional reputation to suggest otherwise.
"The emails obtained by Channel 4 give an insight into the fact that senior editors in BBC News are regularly subjected to lobbying, pressure and complaints from political groupings. It is pressure which it is incumbent on the BBC to resist - and it does. The report showed no evidence that anyone at the BBC acted as a result of these emails.
"The BBC has covered hundreds of stories on Brexit. We have seen no evidence to substantiate claims that our editorial standards have been compromised in any way."