Horner 'completely' denies the allegations, and revealed at the Bahrain Grand Prix that he was 'pleased' that the process was over.
However, the situation doesn't seem to want to go away for the Milton Keynes-based outfit, with a journalist reporting on Thursday that supposed messages allegedly sent by Horner had been emailed to hundreds of accounts by an anonymous source.
Brown calls for F1 and FIA to intervene in Horner process
Now Brown, whose team according to Jenna Fryer was one of the parties who received the apparent WhatsApp messages, has said that Formula 1 and the FIA should conduct an external investigation into the matter.
The American recently said that he believed the allegations to be 'extremely serious', and now believes that a conclusion should be reached by someone other than Red Bull GmbH.
"I read this statement. I think from what I've seen there continues to be a lot of rumours and speculation, questions," Brown said during a press conference after FP1 at the Bahrain Grand Prix.
"I think the sanctioning body has a responsibility and authority to our sport, to our officials. I think all of us in Formula 1 are ambassadors for the sport on and off the track, like you see in other sports, and so I think they need to make sure that things have been fully transparent with them.
"And I don't know what those conversations are and that needs to be thorough, fully transparent, and that they come to the same conclusion that has been given by Red Bull, and that they've agreed with the outcome.
"But I think until then, there'll continue to be speculation because there are a lot of unanswered questions about the whole process.
"So I think that's what's needed by those that run the sport to really be able to draw a line under it. Until then, I think there'll continue to be some level of speculation by people and I don't think that's healthy for the sport."