Stabilise a creature:
Using medicine kit without proficiency in medicine skill
"I got this, I've got a first aid kit!"
This is a DC 10 wisdom check. Add 1 to the DC for negative HP rather than a plain 0, and another 2 for every failed death save so far. Max DC therefore of 15.
Reasoning: possibility of "all the gear, no idea".
Using medicine kit with proficiency in medicine skill
"Stand back, I'm a doctor".
No check required.
Reasoning: they know what they're doing and have the tools they need.
Using medicine skill with no kit available
If proficient: DC 10 wisdom check.
"Stand back, I'm a doctor. The patient needs an IV, stat...oh crap, we're in a dungeon. Here goes!"
Reasoning: they're skilled, but without any equipment, still have a chance of failure.
If not proficient: DC 15 wisdom check with +1 for neg, +2 per failed death save.
"I saw a guy do this one time!"
Reasoning: they're not skilled and have no equipment. It's tricky.
This should make the need for Medicine skill proficiency more appealing. Not to mention the other uses for the skill that others have mentioned, outside of combat stabilisation.