I'm not sure the parallel is exact...
America has the advantage of being much more diverse. Though there was much discrimination and hatred bandied before, during, and after the civil rights movement, there was a large section of Americans that crossed the lines. Indeed, when Medgar Evers and others were killed, they became martyrs for the cause of integration (though in some communities, they probably thought he was an uppity n*gg*r that got what was coming to him) and the perpetrators were widely villanized. I believe that the civil rights movement was successful because America has some sense of justice that can be tapped every once and a while - though it pretty well sits on it's collective arse most of the time.
The main difference, though, in the two battles is one of audience. The drama that was played out in the U.S. in the 50's and 60's had little to do with international press or spin. It was geared mostly at us coming to grips with our own sense of decency and fair play. We had the advantage of having a white class of citizens that did side with the southern blacks (though most were geographically removed). Of course, if the Civil War had ended differently, it would have been an entirely different century.
The tactics in the MidEast, however, have little to do with trying to change the minds of the immediate parties. Instead, there is the effort to get the Superpower(s) and other international concerns to take sides. Problem is, though, that when one doesn't have a direct stake involved, one tends to not pay attention. International politics has a whole different dynamic the national or local politics, and rarely goes beyond self interest.