According to a YouGov poll, even Democrats are more than half again as likely to dislike Islam. Only 17% have a positive view (~2% are themselves Muslims).
I've been thinking about this for a few days, occasioned by the most recent controversy. It seems to me that Islam has both structural and doctrinal commitments that are going to make it problematic for a modern state like the United States, Russia, or any European nation. That's not to say that there are no versions of Islam that are compatible with modern states, to be sure. I mean to say only that it's a harder religion for a modern state to digest.
Structurally, Judaism is universal in the sense that it aims to regulate every aspect of your life. Not every Jew practices it that way (or even close to that way), but if you are ultra-Orthodox, your relationship with God according to the Law will govern everything from what you wear to what you eat to how you pray. Islam is also universal in this way.
But Judaism does not expect non-Jews to live according to the Law: the Law as it envisions it is a part of the special relationship between their nation and the Lord. That others do not do these things is, if anything, a source of pride. Islam does not agree. Legally its stricter forms hold that a few religions (including Judaism) may be tolerated in a subordinate status, provided they accept their submission and pay a tax. Other religions, including animist religions like those common to Africa, or Japanese Shinto, are to be completely suppressed whenever possible.
Structurally, Christianity is universal in the sense that it considers itself to be the one true faith. Islam is like this as well. But Christianity accepts the existence of a secular sphere -- Jesus himself said, "Render unto Caesar" -- and the modern state falls easily into the role he assigned to the Roman empire. Rome can protect the religious liberty of other faiths, and can occupy a space in which many questions are settled otherwise than religiously where religions dispute the proper outcome.
Islam is universal in both the Jewish and the Christian sense. That makes it hard to digest if its adherents take it seriously. In a way, that's a strength of the faith: both Judaism and Christianity have seen many of their mores digested and eliminated by secular Western states. In another way, it's a problem for a modern society. You can't have freedom of conscience if people are free to convert to Islam but not from it on pain of death. You can't have freedom of expression if people are unfree to criticize the faith's leading figures or their doctrines. You can't have a free press that lives in terror of blasphemy laws (again, on pain of death -- not just according to radicals, but apparently according to the inherited law).
These radical interpretations of Islam are not implausible. Indeed, the interpretations given even by nonviolent radical groups -- Hizb-ut Tahrir, say, which claims to be nonviolent but nevertheless finds in Islam a necessary commitment to overthrowing secular states and replacing them with sha'riah -- are not only plausible but obvious. In many cases they are giving the most obvious reading of the tradition.
That it is the most obvious reading doesn't make it the best reading, to say nothing of the only reading. We have the Bible, but also the Summa Theologica. We have a huge tradition of Christian philosophy on how to understand and interpret the Bible. We have the Catechism to help bring those lessons forward in a more understandable way. In Judaism, they have the Torah but also a vast tradition of Rabbinical scholarship. They have a deep, dense, fascinating literature on how to interpret Torah and make it a part of your life.
Islam has a similar road open to it. Indeed, it once had a much more active philosophical tradition. One of the most regularly cited sources in the aforementioned Summa Theologica is Averroes, whom Aquinas calls "the Commentator." It's possible to get there, but the structural and doctrinal issues make it a harder road.
Still, remember these guys. It's a cherry-picked set of examples, sure. But in terms of their numbers, they're no less representative than the radicals are.
America has a right to ask some things of Islam insofar as Muslims would be Americans. Some of those most-obvious interpretations are simply not compatible with the American project. In return, though, a United States Marine with eight tours of duty under his belt has a right to ask some things of America, too.