Outside of explicit contract law you may be tangled up in, the US doesn't have explicit law and most employees are classified as "at-will" (basically, dependent upon where the business views this freedom as beneficial or non-beneficial to itself).
Since businesses haven't managed to lobby such legislation requirements in for workers to give notice (largely because at-will is typically beneficial to them) and just use contract law where it matters (public personalities, etc.) you can just show up and quit legally, without legal repercussions.
With that said, cultural protections have been put into place under job history in the application process. If I were to show up and without a very good reason tell me employer I quit effective immediately, I can do so unlike the European cases described.
But is this really freedom or an illusion of freedom? Many in the US are living paycheck-to-paycheck to survive. They're financially leveraged and can't really just quit, they need to line up an opportunity and have minimal savings to protect them. For them they can make the choice but it's a bad choice for them and most of the time they won't. Businesses know this and leverage it. The pandemic labor market offered a shift in this giving some bargaining power back to labor but it seems to be clawing right back to businesses again, probably for quite some time.
In what are classified as more professional positions, you're expected to give 2-weeks notice. Many employers request references and if you're honest during your interview, leaving without notice in most cases will require a good explanation and a competent interviewer, otherwise you're a red flag no hire almost immediately and there's a pool of others behind you without said red flag. So, I could quit tomorrow, but I'd have to be very careful with my next employer as to how I go about referencing my previous work, why I left, etc. In most cases I can't really just show up and quit, I need to give notice and wait. Perhaps that's not as long as some European countries closer to the order of 4 weeks IIRC but it's not as free as it might appear to be for the majority of laborers.