> Not in the sense that everyone must be the same culture, but the ideals and ethics of the two cultures must be rooted in similar things.

[Citation needed]. The cultures should only be compatible - not necessarily rooted in similar things. Japanese zen-buddhist shame culture is not rooted in similar things as America's typical protestant Christian guilt culture. To my knowledge Japanese immigrants did (and still do) integrate well.

Humans are ridiculously adaptable, I would argue the reverse of your argument - that there is no culture on earth that precludes any immigrant from flourishing in modern, secular western countries. In fact, voluntary immigrants are preselected for willingness to adapt.