Your focus and positive bedside manner further improves the condition of your patients. When you use the Medicine skill to provide first aid, treat deadly wounds, treat disease, or treat poison to another creature, your patient regains stamina points equal to twice your ranks in Medicine. This effect has more to do with the act of giving treatment than the treatment's actual result, allowing the patient to regain stamina points whether or not the associated skill check was a success.
Intention: Create a feat that makes Medicine-based healing worthwhile and cost-effective at low levels without breaking it at high levels or invalidating the Envoy's Inspiring Boost ability.
Too weak? Too good? Just right?
My inclination is "ridiculously overpowered". Its at least as powerful as the relevant Envoy class ability, if not much moreso, at the cost of only a feat. Not only does it scale much faster, but there is no limit on using it multiple times on a given character per 10 minute rest. It also, depending on how you interpret certain rules, can be used multiple times right in a row, because I don't think there is anything in the rules that limits the use of Treat Poison to someone who actually is poisoned.
I envision a really, really cheesy build, where a medic purchases dozens of really really weak poison doses, and uses them as a way to provide functionally infinite stamina recovery outside battle ( or even within battle, if they take the feat that lets them do most of these treatments as a standard action ).
A few considerations:
On The Feat's Power:I think you may have misread this. I specify "ranks" in medicine, not your Medicine bonus (1 rank = 2 Stamina healed). Inspiring Boost restores twice one's envoy level + Cha modifier (eventually scaling up to triple envoy level + Cha mod). It will never heal as much as, let alone more than, Inspiring Boost.
On Limiting Uses: This one is actually interesting. I've been playing under the assumption that the Medicine skill maintained the same retrying rules as Heal from Pathfinder (meaning you could not retry skill checks unless you witnessed evidence of the check's failure), but SF actually does not have this written globally (just Treat Deadly Wounds has a hard per-day limit). Treating disease and poison are actually written in a sort of bizarre way: they say treating takes 10 minutes/a standard actio respectively, but then say "every time the creature attempts a saving throw, you can attempt a Medicine check" with no duration or notes on timing, so I'm not actually sure how that is supposed to work. Hmm.
At any rate, the wording of the Medicine skill is explicit in saying you you can use it on a wounded/poisoned/sick target to do X, as opposed to "you can use it to do X, which has Y effect on wounded/poisoned/sick targets". Based on that wording, the Mecidine skill cannot be used on a healthy person. Moreover, based on the costs of the weakest poisons and medpatches (which are required to use Mecidine as a standard action with the feat you alluded to), assuming you always failed to treat the illness, it would not be cost effective to do so.
tl;dr I need to revise my wording to ensure it can only be used as intended, but when used as intended I don't think it's too powerful.
If you limit it to Treat Deadly Wounds, it avoids the pitfalls of the other uses of medicine and makes more sense (First Aid is done to unconscious creatures after all).
I'd limit it to successful rolls as well, otherwise you can keep recovering stamina points of everybody until you succeed and can’t Treat Deadly Wounds until the next day. I’d scale down to 1 sp per CR/level of the creature, but add the + int mod if you roll over the DC by 5 or more to the stamina gain.
Then it’d turn out like this: can only be used outside of combat, once per day per character (twice with a medlab), requires a medkit or medlab. Pretty much a boost to Treat Deadly Wounds only.
It’s nice to have another alternative to recover Stamina Points other than spending precious resolve points or having an envoy around, and with enough restrictions it can become properly weak enough to be acknowledged as a Starfinder compatible feat.