The US doesn't do exit immigration checks, so if your hypothetical Israeli overstayer can board a flight out of the US, they'll most likely be allowed to go without detention or other punishment. The US sees no value in spending government money to detain someone who is ready to leave of their own accord.
I'm not sure why such a person should have "no valid documentation". Maybe you're confused by the common term "undocumented" for people without legal immigration status, but it usually doesn't literally mean they have no documentation at all. Most such people would still have their passport and other identification from their home country, and the passport is enough to board a flight back home.
You say your hypothetical Israeli overstayer had a visa; so they must have had an Israeli passport when they entered the US. Under normal circumstances, they've still got it. If it expired, they could have renewed it at an Israeli consulate within the US; the consulate will issue passports to all Israeli citizens without regard to their immigration status in the US. Even if somehow they lost their Israeli passport and all other identity documents, the consulate would have the ability to verify their identity from Israeli government records (e.g. the photo and biometrics from their previous passport, which would still be in the Israeli government's database), and process their application for a replacement passport.