Red Bull reached a settlement with the FIA over its £1.8 million breach of last year's budget cap ahead of the Mexican Grand Prix, arguing that it misinterpreted some of the permitted exclusions.
The team was hit with a $7m fine and a 10% reduction in its permitted aerodynamic development for the next 12 months, a sanction Horner called "enormous" and "draconian."
It brought an end to the cost cap saga that first emerged over the Singapore Grand Prix weekend in late September, when the first whispers emerged that Red Bull had breached the budget cap.
Horner hit back emphatically against the accusations when they were first made, as well as calling out rivals for making "defamatory" claims without any official information from the FIA.
Red Bull was only officially informed that it had breached the cost cap on the Sunday of the Japanese Grand Prix, 90 minutes after Max Verstappen clinched his second world title. A public notice was then issued one day later.
Reflecting on the case in Mexico following the FIA's ruling, Horner said he expected follow-up action to investigate why news of Red Bull's breach had leaked out.
"The accusations made in Singapore were extremely upsetting for every single member of staff, all our partners, everyone involved within Red Bull," said Horner.
"Obviously, any form of leakage is hugely worrying. It's something that we expect to be followed up."
Prior to the Accepted Breach Agreement being struck between Red Bull and the FIA, a number of the team's rivals had been clear in the calls for strong action.
McLaren boss Zak Brown wrote to FIA president Mohammed Ben Sulayem and F1 president Stefano Domenicali to say that any possible breach of the cost cap "constitutes cheating", something Horner fiercely hit back against in Austin.
Horner said after the FIA's ruling that he felt Red Bull was "probably due an apology from some of our rivals for some of the claims that they've made," and that the team would "make no apology for the way that we've performed, the way that we've acted."
Horner added: "We do take on the chin that there are lessons to be made. Potentially mistakes have been made in our submission, which with the benefit of hindsight and 20/20 vision, everybody can be a specialist.
"But there was no intent, there was nothing dishonest, and there was certainly no cheating involved which has been alleged in certain quarters. So I don't feel that we need to apologise.
"I think there are lessons that have been learned. Everybody can learn from this. We've taken our pounding in public, we've taken a very public pounding, through the accusations that have been made by other teams.
"Our drivers have been booed at circuits. And the reputational damage that has been made by allegations has been significant. The time has come for that to stop."
