The Republicans supported the Civil rights bill. The racist democrats, known as the Dixiecrats at the time, wanted the bill defeated.
Yes, the white Republicans supported the movement. The white democrats did not. (The left wing powers that be have berried this fact from the American publics conciseness. Heck, Byrd was a grand poomba of the KKK).
When in the military....years ago....I was stationed in the deep south. Outside the base was a small town that was extremely racist. The military was very careful about not stationing blacks on that base, although we did have some that seldom ever left the base. One night I was ordered to pick up an officer at the train depot. It was the middle of the night. He wanted some coffee and the only place open was the Greyhound bus station. This office knew nothing about racism in this part of the world. A black walked down the sidewalk. When he approached us, he went into the street to go around us. The officer told him he didn’ have to do that. He offered to buy him some coffee in the bus station. I told him to “let him be”. “If he went into that bus station and sat with us, you will get him killed.” That’s not right but that’s the way it was. The point being is that this officer had no idea that this was going on there. Whites living out of the area had no clue as to how the deep south treated blacks. Even I was amazed....being from the west coast. As a 20 year old back then, it was a rude awakening.