Well, almost. Bob'sWorld said:
"Americans take an insanely conflicted position on Islam. On one hand we blame the 9/11 attacks and the GWOT in general on Islam ("Extremist Islamic Ideology is the strategic center of gravity"), on the other hand we bend over backward to accomodate our Western perceptions of Islam as we enter their lands to conduct military operations (policies on alcohol, mosques, etc)."
I read similar things in the papers and on the internet but I travel a fair amount and no one I actually talk to expresses such sentiments. Either direction. Unless they're repeating the media or some talking head (which contrary to said media and Hollywood, very few Americans actually do. Fortunately). Few are inclined to blame Islam for all things or to say that Islamist ideology is a or the center of gravity -- that, BTW, is a much overused and abused term; Center of Gravity, I mean -- nor do I run across many willing to give Islam a pass on some of their more inane mutterings.

Thus I'm not sure most Americans are particularly conflicted about Islam. Some are, no question but most, I don't think so. Thus that would appear to me to be a matter of context as in who expresses such sentiments or conflicts -- and why...

Rex said:
"Similarly, most Americans would be happy to invite a Muslim to their local church—but presumably would get rather more upset if it was (say) Iranian combat troops searching for American "resistance fighters."
Possibly. My guess would be about 5-10% would be truly upset on the religious angle, about the same amount would be angry over the military angle and another 40-60% would yell about it just for the heck of it and because it would be something to gripe about and perhaps garner some sympathy.

Like you, I've wandered in and out of mosques all over the ME with nary a problem. That leads me to believe that my guesstimate for annoyed Americans in your scenario is a fairly close match to the percentages of Muslims and their reasons for the protests over the 'violation of mosques' by infidels...

As you say, context is important...