The short answer is "no".
Tracking along a DME arc requires that you first intercept it. That point of interception is normally shown as an IAF and the fix requires an azimuth navaid as well as DME, in this instance. The Localiser radiates in one direction only, so you need either a VOR or an NDB, to provide track guidance to the IAF and, therefore, to properly identify the fix.
Then it's a good idea to have an azimuth, so that you can monitor your progress around the arc. It becomes even more important for identifying the point at which to turn off the arc. A Localiser signal can't do any of that for you.