That was the original assumption. However, basic [0] orbital mechanics says that objects with smaller orbits orbit faster. If this were applied to the galaxies spiral arms, then as they continue to orbit, the inside ones would orbit faster than the outside ones, and the galaxy would become increasingly spirally. By this point, the galaxy would be so tightly wound that we would not even be able to see the spiral any more.
A more modern hypothesis is that the arms are more like waves of high density. Individual stars move through the the arm.
Since this high density region exerts gravity, stars that are moving out of it are slowed down, and stars that are moving into it are sped up. The end result is that orbiting stars tend to spend a little bit more time in the arm region then they would under a pure orbit. In doing so, they contribute to the mass of the arm, causing other stars to also spend more time in the arm then they otherwise would. The end result is a stable wave as all the individual stars move into and out of it throughout their orbits.
https://en.wikipedia.org/wiki/Density_wave_theory
[0] As in, just applying Newtonian gravity.