What's the best way to heal a nasty abrasion that never bleeds?

Neosporin or Bacitacin are good. Especially right out of the shower when it is at its raw phase. Do this a few times a day and that should clear it up. Keep it uncovered as much as possible as a moist dark spot grows bacteria. Do not make it bleed!!! This could lead to a staph infection such as MRSA (an antibiotic resistant staph infection). Watch for signs of any pus, odor, increase redness around the area, swelling and increase in warmth. But other than that you should be ok.