At the time, region predicted support more than party: northern Democrats and Republicans voted for the civil rights act, while southern Democrats and Republicans voted against it. With a handful of exceptions on both sides.
That is likely true, but the common claim is that the entirety of the Republicans are the "racist" party. This requires that all Republicans suddenly decided to reverse themselves within a short time period. Your statement is probably the closest to what was going on at the time instead of the usual blanket generalizations that skip over the facts of the era.
