Since alpha-particles (used in Rutherford's experiment) have positive charge, they probably should attract electrons (and stop being alpha particles). Was it the case in Rutherford's experiment? Did the foil become slightly positively charged (ionized)?
The phenomenon that occurs in the experiment is known as Rutherford Scattering. That second word is rather important - the $\alpha$-particles scatter and don't stop in the gold foil. You start with a tightly collimated beam of $\alpha$s and observe that, after you put the foil in the path, the beam gets slightly diffused in the forward direction. What is surprising is that you get a significant amount of back-scatter - that is, $\alpha$s coming back from the foil.
In either event, the $\alpha$s don't stay in the foil so it doesn't "charge up".
For the pedants: That's not to say there isn't any local ionisation. An $\alpha$-particle is highly ionising and will strip electrons readily from atoms in the foil as it passes through.