I guess you haven't looked at the actual record, but I am not the one who is rewriting history here. As a percentage of their respective members, more Republicans voted NO on the 1964 Civil Rights Act than Democrats. Not one single southern Republican member of the House or the Senate voted yes. None. Yet northern Democrats even voted more in favor than northern Republicans. There were more House members from northern states who were Republicans (162) than Democrats (154), yet more Democrats voted yes than Republicans (145 to 138). 

So yes, given the split in Congress, Republican votes were needed to pass it, but more Republicans voted no, as a proportion of their members, and more Democrats voted yes.