Let us be clear the President of the United States is not a stupid man who speaks without thinking and when he speaks of "we", unless he makes it clear he is talking about some specific group like the Democratic Party or his family, he is speaking of the American people. So when he says
"The idea that we would burn the sacred texts of someone else's religion is contrary to what this country stands for,...It's contrary to what this nation was founded on, and my hope is that this individual prays on it and refrains from doing it."he is clearly stating that Islam is "someone else's religion", not the religion of Americans and thus that Muslims are not real Americans. Indeed he is accepting the religious right's assertion that, despite what the Constitution of the United States says and what the founding fathers intended, the United States is a Christian country and Americans are Christians.
This is the almost inescapable result of the American attempt to take a nation of immigrants from many different countries and create an artificial ethnic nation from it. For some people it may be easy, just change your last name, for others it is not so easy to change the colour of your skin or indeed the religion of your birth or choice if that is required to be considered real Americans.
I must say, I much prefer the inclusive Canadian multiculturalism where our nationality is not based on ethnicity, but on citizenship.