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TIL notarial acts in Dutch law.
As for whether the concept exists directly in Anglo-American law, I don't think there's a direct analog, but some procedures in contract and property law can get close.
The US Constitution's Case or Controversy clause means that judicial cases cannot be adjudicated in advance of an actual dispute, which would generally mean that judicial enforcement (like garnishing wages or seizing assets) is not yet available. The same generally applies in minor criminal cases or infractions, so things like on-the-spot penalties for roadway offenses are not a thing.
However, if we broadly consider the idea of "a document that is taken at face value" once signed and without the need for vindication through the courts, then perhaps some documents in US law will meet that definition. For example, US judges will often issue forward-dated orders that take effect upon a specific condition, on a specific date. Such orders are fully valid, and failure to comply becomes a criminal offense. However, this is kinda obvious, that a court's order would be recognized as mandatory.
If we look at private parties, such valid-once-signed documents are rarer, but not non-existent. One can argue that deed restrictions on property meet that criteria, with a common example being an HOA restriction on real property. Repeat violation of HOA rules can allow the HOA to assess penalties against the homeowner, with the highest penalty being a forced sale and seizure of the property. Whereas typical contract law would permit the target of such action to petition a court to intervene in advance, the juncture of contract and property law may give deed restrictions different treatment, depending on the US State in question. It is plausible that some jurisdiction will allow collection of the supposed penalty first, and only allow contesting the penalty afterwards. Functionally, this could be viewed as a "notarial act" but judicial review available.
One other example might be the Payable On Death (POD) beneficiary designation for bank accounts. Authorized by most US States (except Texas and Louisiana, apparently), such model legislation allows for a bank's recognition of POD beneficiaries upon the owner's death, without going through the normal procedure of legal probate (to divide the owner's estate). This essentially means that this portion of the contract between the account owner and the bank is self-executing. That said, the bank would need to first recognize that the owner is dead, and that usually requires a signed death certificate, issued by a coroner or judge. So maybe not totally self executing, but the triggering condition would set into motion an action which does not require judicial intervention to implement.