Of course not.
Biden is not the Vice President of Barack Obama.
He is the Vice President of the United States, an independent and separate Constitutional office. He was elected to that position legally, in accord with Article II and Amendment XII, and in the case of a vacancy in the Office of President, FOR ANY REASON AND OF WHATEVER CAUSE, Biden becomes President.
EVen an illeagal election?
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You know, at first blush I agree with you except for one thing. Consider this:
The Electoral College votes (EC) and the results go to the House where they are charged with determining th eligibility of the candidate who wins. The rules there specifically state (I'll find them if I must) that if the House finds that the winner is ineligible the House declares the 2nd place vote getter to be President.
Now it use to be 2nd place was named Vice President anyway. But now, Vice President isn't a separate ticket, they are together. Second place would have been Mccain/Palin. So there is an argument that the election would go back to the House where they would be compelled to unravel the election, and declare the 2nd place to be President.
I'm not saying you are wrong, just that there is specific language in the rules for just such an occurrence. What isn't contemplated is what happens if the House doesn't catch the ineligibility and it is caught later. But still I think Constitutionally it would fall back to the House to fix. I would expect them to try to declare Biden to be President, but I don't know.