Horner said the insinuation has led to children of Red Bull staff being bullied at school.
Horner sat alongside Brown in a fiery press conference supposed to preview Sunday's United States Grand Prix - the first race since Max Verstappen claimed his second world championship - but it instead centred around the cost cap row which has engulfed Formula One.
Red Bull exceeded last year's £114million budget cap when they carried Verstappen to his contentious title win over Lewis Hamilton.
Although F1's governing body the FIA is yet to confirm by how much Red Bull overspent, it is understood the figure is £1.8m.
Earlier this week, Brown wrote to FIA president Mohammed Ben Sulayem and said a financial breach "constitutes cheating".
"Obviously Zak's letter is tremendously disappointing," said Red Bull team principal Horner.
"For a fellow competitor to accuse you of cheating, and to accuse you of fraudulent activity is shocking.
"It is absolutely shocking that another competitor, without the facts or without any knowledge of the details, can be making those kind of accusations.
"We have been on trial because of public accusations since the Singapore Grand Prix (on October 2) with the rhetoric of cheating, and the rhetoric that we have had this enormous benefit.
"Numbers have been put out in the media that are miles out of reality and that damages the brand, our partners, our drivers, and our workforce.
"In an age where mental health is prevalent, we are seeing significant issues within our workforce.
"Kids are being bullied in playgrounds because they are children of Red Bull employees. That is not right and it has come about because of fictious allegations from other teams without any fact or substance.
"We are absolutely appalled at that behaviour. We have been subjected to three weeks of effective abuse. It is just not right and it has to stop."
Red Bull have been offered an Accepted Breach Agreement by the FIA, with both financial and sporting sanctions believed to be on the table.
Such a deal, which has involved back-and-forth negotiations between Horner and Ben Sulayem here in Austin, would see Red Bull lose any right to appeal.
Red Bull could yet challenge the FIA's decision. Their case would then be heard by a panel of independent judges, while a final option would be the FIA's International Court of Appeal.
However, both steps would see the team risk a harsher punishment - with the deduction of last season's constructors' points possible - and ensure the row plays out for several months.
Horner added: "I hoped it would be resolved before this weekend and I am hopeful it will be resolved during this weekend.
"But should that not happen, the next process is a cost cap administration panel and then beyond that there is an International Court of Appeal so it could be drawn out for another six to nine months which is not our intention. We want closure on 2021. I am hopeful of a conclusion in the near future."
Red Bull are believed to have put the overspend down to internal costs related to sick pay and gardening leave and a change as to how unused spare parts are allocated.
"We had zero benefit from a development perspective or an operational perspective for 2021 or 2022 from the way we operated within the cap," concluded Horner.
"We absolutely and categorically don't feel we have had any advantage. It is totally fictitious.
"Once this situation has been concluded I will talk you through the reasoning behind our submission and our position, and why we believe there is a contrary position. The whole thing should be transparent. There is going to be no private, secret deal. It will absolutely be above board."