Uh... when did Trump say this?
No, seriously, when? In all the many attacks on Trump, I haven't heard of this before. I tried searching but couldn't find anything. If Trump said "Obama can't be an American because he's black" the liberals would be all over this, but I've never heard of this.
I'm guessing this is a reference to how Trump for a while was promoting the "birther" conspiracy that Obama wasn't born in the United States and as a result wasn't eligible for being President. That's certainly a silly idea, but that's a pretty far cry from "Obama can't be an American because he's black".
There is little reason to believe that Musk was doing a Nazi salute; he even preceded it with saying "my heart goes out to you" to better show the purpose behind it.
In some fairness, Musk's handling of the accusation afterwards was bad. But I don't see much reason to believe it was an intentional Nazi salute.
Who?
I'm guessing this is in reference to Marko Elez, who was hired for DOGE but then resigned after it was discovered he made a bunch of racist posts on a separate account under a different name. I don't know if there's any reason to believe Musk was aware of that at the time of hiring, but I suppose that point doesn't matter anymore because he was then re-hired by Musk (or at least Musk said he'd re-hire him recently, I'm not sure if he actually did yet).
Regardless, even if tweets like "Normalize Indian hate" are clearly racist, I haven't seen any evidence he was a Nazi supporter, despite having looked at several liberal-aligned articles complaining about him and sharing examples of the racist tweets. Surely if there was any actual evidence of Nazi support, they'd be showing that... but no, it's just generic racist stuff. Obviously not good, but not Nazi either.