The gay rights movement IOTL happened as a result of two major social influences--a reaction against the anti-homosexual culture beforehand, fostered by the growing secularization of the West, and the rise of the Feminist movement. The latter stimulated gay rights on the basis of the feminist advocacy of 'sexual liberation.'
Asexuality, frankly, has never had the degree of social stigma that homosexuality has. In fact, I'd venture so far as to say that asexuality was considered a desirable trait before the Sexual Revolution. Consider the Victorians, who worked hard to keep sexuality in general behind a locked door. Consider the overall tradition of Western philosophy, which, especially since the rise of the organized Christian Church, has advocated a certain godliness, a detachment from the affairs of the world, and, starting with the Enlightenment, a separation from the rather base desires that characterize sexuality. There was always a social niche for asexuals--the monastery, the convent, the laboratory, the study.
To echo what the others in this thread have said, an asexual pride movement would require a backlash against an earlier, excessively-sexual society, or a greater philosophical movement toward the concept of the New Man, who is rational enough to not have sexual desire.