Red Bull's parent company in Austria announced in a statement on Wednesday that the grievance against Horner, who denied the allegations, had been dismissed after a lengthy independent investigation.
Red Bull have said the investigation report will remain confidential.
"I think the sanctioning body has a responsibility and authority to our sport, to our fans," Brown told reporters in a press conference organised by the governing FIA at the season-opening Bahrain Grand Prix on Thursday.
"So I think they need to make sure that things have been fully transparent with them.
"That’s what’s needed by those who run the sport to be really 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."
Wolff, whose rivalry with Horner has provided much fodder for Netflix docu-series Drive to Survive, also emphasised the need for transparency.
"Eventually things are going to happen, and I think we have the duty, the organisation has the duty to say well we’ve looked at it, and it’s OK, and then we can move on," said the Austrian.
Horner spoke to Sky Sports television briefly, as he walked through the Sakhir paddock, and said he was pleased the process was over.
"I obviously can't comment about it but here, very much the focus now on the grand prix and the season ahead and trying to defend both of our titles," he said.
Key paddock figures and permanently accredited journalists later received an email from an anonymous sender containing a link to material purported to be evidence submitted during the investigation.
"I won’t comment on anonymous speculation," Horner said in a subsequent statement.
"But to reiterate I have always denied the allegations. I respected the integrity of the independent investigation and fully cooperated with it every step of the way.
"It was a thorough and fair investigation conducted by an independent specialist barrister and it has concluded dismissing the complaint made. I remain fully focused on the start of the season."
