If just the nail snapped there's no reason for it to bleed. The only time a nail will bleed is if it's cut or torn into the quick - just like yours.
You'll want to get a set of clippers for your dog's nails and clip them regularly. Some people use a dremel file. Either way, you just have to watch and not get into the pink "toe" inside the nail. It's a good idea to give the nail a few final swipes with a nail file if you use the clippers. It will smooth off the rough edges and help keep them from snagging and tearing.