My dog's nipples are red, itchy, with black spots. Vet or home care?

That looks uncomfortable. It's impossible to know at this stage if the nipple is infected or if the skin on/around the nipple is inflamed and this is causing the enlarged size. The block dots could be flea dirt, blocked pores or debris Patrick has laid in.
Your best option is to have Patrick examined by the veterinarian and to get medications to help the itchiness. These lesions could be several things including fleas, allergies, skin mites, bacterial infection, ringworm (which humans can get) or endocrine disease.
If you can't get him in to the veterinarian, there are a couple things you can do at home and they can help but they may not address the underlying cause. You can try adding in an omega 3 fatty acid supplement for dogs, this can help improve the skin health. You may also try changing his diet to something like Royal Canin or Nutro as they are aimed at skin health. You can wash the lesions once a day (not his whole body, just the lesions). Also protect the area from trauma by having your pup wear an elizabethan collar at all times. If the lesions do not start improving within a week or if they are getting worse, have him examined by the veterinarian. I hope this helps and please feel free to post any additional questions.