Not on your life. They may grumble a bit, and go grudgingly to the table in frustration the next time they talk about in-church things, a.k.a. ecumenism. But even there, it is not likely that the Orthodox and many kinds of Protestants, who claim that they are the original Christian church from whom all others deviate, will do more than sulk. Harvard theologian Harvey Cox put the matter in perspective: "We" knew all along what official Catholicism taught, but we took it with a grain of salt. This month's statement issued by the Congregation for the Doctrine of the Faith only calls forth larger grains of more salt. Otherwise, no problem. They'll keep talking.
Intrachurch and ecumenical Christian life is the subject of our "Sightings" only when we see it directly impinging upon extrachurch life, the public order. The statement may sound abrupt and retrogressive to non-Catholics: It steps back from the Second Vatican Council, which called them "brothers," and from the writings of Pope Paul VI, who called them "sisters," to demand that Catholicism now be thought of as the "mother" and all the others as "daughters." But most non-Catholics know that this was aimed more at relativizing Catholics than at anyone else. So everyone else will go along as before.
How have Catholics and non-Catholics lived both with the notion that each is the "full" or "true" church and that salvation goes on beyond the scope of each? Are they all indeed relativists? Something more subtle than that goes on. Historian Sidney E. Mead made much of a report in 1972, in which three-fourths of young people among the doctrinally rigid Missouri Synod Lutherans told poll-takers that they believed both that "belief in Jesus Christ is absolutely necessary for salvation" and that "all religions lead to the same God." These are "two theologically contrary views," wrote Mead, yet they did not paralyze anyone.
Do your own poll-taking and you will find that however exclusivist and absolutist Catholics or members of other communions are, most clergy and lay people think something bigger is also going on in, through, around, and beyond each. No statement from the headquarters of any group will tidy up such contradictions, which are so close to the center of American public religion. And private, too. They will work together, as before.