I would apply the same rules to the USA yes, but the way you summarized it is **not** what I suggested for israel. Again no one is required to leave.
If it were to be applied to the USA then the US government would dissolve and the native american government would be the default. All people who choose to stay born on the land would become citizens and all people not born on the land may be allowed to stay at the discretion of the new government.
Since we would all be citizens in the new government we could all vote collectively on how or if we want people not born on the land to be treated and if they will be granted residence or not. It is very likely such a vote (Which would largely be non-natives as natives are a small portion of the population) would allow the immigrants to stay.
In the case of israel the same would happen but since the palestinians are in much greater numbers the vote in the new unified nation would be more fairly split between the two, and thus would have some concern for the jews who remain baked in as a result, since they are a huge portion of the vote.