It does, but only if they can come here and take the money. (See https://en.wikipedia.org/wiki/State_immunity , https://en.wikipedia.org/wiki/Sovereign_immunity_in_the_Unit... , and https://en.wikipedia.org/wiki/Foreign_Sovereign_Immunities_A... for some of the underlying concepts here.) I don't expect many countries to start suing the US -- good diplomatic and trade relations are too important.

Saudi Arabia is in a unique position here: they're dependent on the US for defense, the royal family has a lot of Salafi and al-Qaida sympathizers (including people whose donations to really sketchy Islamic charities went to al-Qaida; I've heard that the director of one such charity bought bin Laden his satellite phone). So, we can sue them and make them comply with our terms; if they choose to exercise state immunity to exempt themselves from 9/11 lawsuits, we can compel them to change their minds, by force of arms if necessary.

(By the way, Saudi Arabia is why mainstream opinion is horrified at Trump's idea of letting the Japanese have nuclear weapons. Japan can be trusted with nuclear weapons, for obvious reasons, but Saudi Arabia can't; and if we let the Japanese get the bomb, the Saudis will expect us to let them have it, too. At that point, what argument can we make? If we just came out and told them that they aren't grown-ups yet, they'd probably throw a tantrum.)

I can't tell if you're genuinely asking or just making a snarky point, but no. It's a US Law pertaining to "foreign" governments, so I don't see how it could apply to the US Government. And it will be vetoed anyway.