I don't think the opposition was as widespread as you believe

In the months leading up to the bill being signed on July 2, there was support for the act, but still a third opposed the bill. One month after its passage, when the implementation phase began, support was just more than 50%, with nearly 1 in five voicing uncertainty about the bill. 

Source http://ropercenter.cornell.edu/public-opinion-on-civil-rights-reflections-on-the-civil-rights