Please, please, please, folks do NOT soak arterial wounds or even venous wounds in water, not hot or cold or ice water. Not with man nor beast.
If the dog stopped bleeding with its foot in water that is a lucky coincidence and nothing more. Hopefully the water did not get down to the base of the wound and it was able therefore to clot. It is NOT how wounds that continue to bleed aggressively should be dealt with.
It prevents blood clotting and you can cause an animal to bleed to death. ANY amount of moisture around a wound can interfere with clotting.
Especially when soaking in water the amount of blood being lost is not apparent; that is an additional issue. An animal can bleed to death before the amount of blood loss is apparent.
Ice and cold will NOT cause a spurting wound to clot, and if enough cold is used that circulation is actually impaired, you will cause serious tissue damage - frostbite, which may in turn require amputation or euthenasia.
Additionally, keep in mind that with dogs, their circulation in their extremities is extremely powerful, and they have normal circulation even when walking on ice or wading through chest deep snow. Sufficient ice to slow circulation WILL call cold damage (frostbite). Pad and foot wounds need to be bandaged tightly, with more wraps added over and the original not removed, and the dog needs to go to the vet for serious wounds that continue to bleed.
Please apply pressure and leave the pressure bandage on. Do not apply so much pressure that circulation to the lower extremity is reduced - continue to feel for a pulse and a continuing normal level of warmth in the extremity periodically.
Do not put anything in the wound to 'clot' the blood - it will only encourage infection and have to be removed in order to debride the wound and keep it draining normally. Especially with punctures they must heal from the bottom up - if you try to close the top of the wound it WILL VERY LIKELY become infected.