"Western culture" already has had many contacts with Islam. "Algebra", "assassin", "admiral, "albatross", "adobe"... there have been extensive interactions between the Occidental and the Oriental world, and not all have ended in gunfire.
"Western culture" has assimilated faiths and coexisted with people ranging from Native Americans to African-Americans. In the United States, we have people practicing Judaism, Islam, Hinduism, Buddhism, Santeria, forms of Voodoo, Rastafari... Just because the dominant culture is Christian doesn't make our institutions only work with Christians.
Islam is not just Wahhabism. It is also the Sufis, the Arabian Nights, the rich cultures of a beautiful region. I suggest anyone who has doubts that the faith of Mohammed can coexist with any society read the stories about Nasruddin Hodja.
The good aspects of Western culture are derived from periods of enlightenment. From the best of the Renaissance and the Enlightenment to the best of the Romantics, Western societies say that every human being has rights and every human being has something to contribute. Everyone is equal before the law and everyone has the right to participate. People have the right to seek out a better life, to work hard, and to trade with others. Their faith is irrelevant. Their ethno-racial group is irrelevant. And while a culture with totalitarian values may not be consistent with Western societies (though frankly I would suggest that people look carefully at Christian fundamentalists, right-wing parties in Europe and fascists throughout European history to see how consistent Western culture often empirically is with violence and totalitarian ideologies), many other cultures that are different from the majority are.
It is perfectly possible for Muslims to live alongside non-Muslims, just like it's possible for Christians, Jews, atheists, agnostics, Buddhists and neo-Pagans to get along.