Do supermassive black holes cause galaxy rotation?
Not in and of themselves, no.
Part of the problem with answering this is that we have no idea how galaxies formed in the first place, none are forming right now, as far as we know.
But we do know a lot of how a star is formed. And using that we can postulate about galaxies. A star forms from pools of gas that start clinging together. As their mass increases, a "dent" in the space/time continuum forms which we know as gravity. As the gravity increases it pulls in more hydrogen but here Newton's laws of motion are in play, particularly the 'body in motion tends to stay in motion and in a straight line.' The force of gravity pulls the atom out of its straight light motion. You can observe this in your own bathtub when you drain it. The water does not fall straight down the pipe but swirls around it.
This same principle is at work in galaxies.