No, it is pretty random, just as the orbits of binary stars are randomly oriented. The scales of star and solar-system formation are much, much smaller than the scale of the galaxy, so the memory of the latter has been scrambled by all the intermediate and local processes.

From this Cornell "Ask an Astronomer" page (http://curious.astro.cornell.edu/about-us/159-our-solar-syst...)

"The size of a solar system is so much smaller than the size of the Galaxy, that the Galaxy's structure has no impact on the orientation of a solar system. What determines their orientations is the direction of the angular momentum that the system had when it formed, and that's pretty much random."

Or take a look at these ALMA images of protoplanetary disks (solar systems in the making) around pre-main-sequence stars. All are at very similar distances, and most are at very similar positions relative to the Galactic plane, so they should all show very similar projected shapes and orientations if they followed the galaxy's orientation -- but they are all over the place.

https://www.eso.org/public/images/potw1904a/