Can Fraxel Repair Permanently Remove Milia?

Fraxel is not necessarily the best way to treat milia.
In fact, this has the potential to cause milia.
Any penetrating injury can be a setup for mila formation.

Hopefully, you are not undergoing something as expensive as Fraxel to remove milia. I suppose as a side benefit, Fraxel will probably remove many milia but there certainly are other, less intrusive and cheaper methods such as extraction. It would be more prudent to wait for a dermatology problem and have your dermatologist remove them gratis.
As for permanence, Fraxel, extractions and dreams can not prevent them from recurring. Using a Retinoid ( such as Retin A, Tazarac, or Differin) or an alpha or beta hydroxy acid would help. Also, making sure your makeup is is not too occlusive will help ( powders preferred over thicker liquid). Do not use anything that will abrade or irritate the skin such as a strong exfoliant. Sunscreen also will help cut down on milia especially in the areas lateral to and below your eyes. .