Anyone who comes to this country or any other country, from another place, is an immigrant, that is just a fact, it has no cultural or moral meaning, it just is.
It is the obligation of the immigrant to accept the values and culture of the country they have come to live in. If I went to an Islamic country, I would show respect for their culture and values, this doesn't mean I will change my own values or beliefs, it means that I would respect theirs, while I was in their country.
I do think we can reasonably expect the same from people who wish to immigrate to our country. If they are not willing to do this, they should be denied entry. In particular immigrants need to respect and follow the laws of their country of choice, or suffer the consequences of not doing so.