It depends on how the authorities want to treat the case, and what kind of act we’re talking about. Within the Westerosi tradition, we know that Eddard Stark as Hand of the King declared Gregor Clegane’s attack on the Riverlands worthy of attainder, which is associated with serious felonies and treason, and that historically kings like Aegon V have dealt with breaches of the peace by leading royal armies in the field to arrest and quite likely execute the violators.
On the other hand, in the English legal tradition, making a “breach of the peace,” wasn’t technically a criminal or civil offense, because it’s considered a violation of the royal prerogative to maintain the peace. So what happens when you commit a breach of the peace is that you get arrested, put in front of a magistrate, and “bound over” (essentially put on probation/bail) where you are required to refrain from certain activities (usually but not always the activity that involved the breach) for a given period of time. However, if you violate the terms of your binding, you’re in contempt of court, and then criminal penalties involve.
However, a lot depends on what kind of act we’re talking about. A drunken fistfight is technically a breach of the peace, but so is banditry. And given the context of Ser Gregor Clegane’s case, he was being accused of banditry, murder, wanton destruction of property, and in short being an “outlaw.” And in early common law, if you defied the laws of the realm you could be declared an outlaw or a legal non-person – which meant that murdering you was legal, and helping you was a crime. It’s arguably worse than being a traitor, as legal penalties go.
Likewise, if the powers that be decided that your breach of the peace was made as a deliberate insult towards the king – that you were implying through your action that the king was too weak or feeble to defend the peace, for example – you might be guilty of Lèse-majesté, and that is a form of treason.