Here is an example of the type of situation that brought up this question. Say a PC has been pinned under the foot of a massive animated stone statue. The PC in question is knocked unconscious by the initial damage and will take additional damage each round, forcing failed death saves, as the statue continues to crush them. A nearby ally succeeds at a Medicine check to stabilize the pinned character so they won't have to make a Death Saving Throw on their next turn. On their next turn the PC being crushed will take a small amount, and suffer a failed death save. An ally will then make another Medicine check, making them stable again. Rinse and repeat.
Under Stabilize A Creature in the 5E PHB it states
If healing is unavailable, the creature can at least be stabilized so that it isn't killed by a failed death saving throw.
This means that with a DC 10 Wisdom (Medicine) check you can stabilize someone even if they have already failed one or two death saving throws. There seems to be nothing to prevent an ally, or allies if the first one fails their Medicine check, from indefinitely eliminating the risk of death in this circumstance with ongoing first aid.
Can Medicine checks be used, with decent rolls, to completely mitigate the risk of death from ongoing damage?