The long and short of it is that it is totally fine for your dog to eat ice. There is nothing inherently dangerous about your dog chewing on ice. Many dogs think that ice is a fine treat. The bigger issue is cooling your dog down too quickly.
If your pet has become over heated, giving them ice or ice water is not a good idea. Letting an overheated dog sit in ice water will cause the capillaries on their skin to close and prevent their internal organs from cooling down. If your dog is hot it is best to cool them down with a damp towel. Once they start to cool off, give them some cool water to drink.
If your dog is too hot and is drinking too fast they may ingest a lot of air. This can cause bloating in their stomach which may cause some stomach upset or even a twisted gut. If they are drinking too fast, remove the water and help them cool down by walking or with a cool wet towel.
As far as giving your dog ice, that is fine. The only danger is to their teeth. Chewing on ice can cause their teeth to chip, so keep an eye out for that. So if your pup find a nice icicle, as long as it is clean, go ahead and let him chew on it. It could be a cold but fun treat.