The rules say

Herbalism Kit. This kit contains a variety of instruments such as clippers, mortar and pestle, and pouches and vials used by herbalists to create remedies and potions. Proficiency with this kit lets you add your proficiency bonus to any ability checks you make to identify or apply herbs. Also, proficiency with this kit is required to create antitoxin and potions of healing.

And

You can use your action to administer first aid to an unconscious creature and attempt to stabilize it, which requires a successful DC 10 Wisdom (Medicine) check.

So there is no direct link and my initial reaction was to say "no", however , the rules also say

2. The players describe what they want to do.

If a player says "I stabalise using a poultice and smelling salts" I would be inclined to reward the inventiveness.

The rules say

Herbalism Kit. This kit contains a variety of instruments such as clippers, mortar and pestle, and pouches and vials used by herbalists to create remedies and potions. Proficiency with this kit lets you add your proficiency bonus to any ability checks you make to identify or apply herbs. Also, proficiency with this kit is required to create antitoxin and potions of healing.

And

You can use your action to administer first aid to an unconscious creature and attempt to stabilize it, which requires a successful DC 10 Wisdom (Medicine) check.

So there is no direct link and my initial reaction was to say "no", however , the rules also say

2. The players describe what they want to do.

If a player says "I stabalise using a poultice and smelling salts" I would be inclined to reward the inventiveness.

Of course, you can only get proficiency once so no point if you already have proficiency in medicine.