Your DM would have to make a house rule.
A character with Medicine proficiency should in theory be able to help another character more easily recover from injury, disease, or poison. There are specific injuries that do call for a Wisdom (Medicine) check, but I haven't found any RAW mechanics in this edition for using Medicine to end disease or the poisoned condition. How Medicine checks can "treat a patient" would probably require a house rule from the DM, and then that rule would have to be in place to see how it affects Recuperating.
The citations that follow are based the premise that the effects where Medicine might help have both an initial Constitution saving throw where failure imposes a condition, and subsequent Constitution saving throws to recover. If Medicine boosts or replaces some of these saving throws, most of the effects under consideration need to match that description.
Other than stabilizing the dying or diagnosing an illness, using Medicine is scarcely mentioned in the PHB and infrequently mentioned in the other core books.
Under Adventurer's League rules, you could treat a patient for injury, disease, or poisoning with Keoghtom's Ointment which can address any or all of these. But that's not using the Medicine skill, now is it? AL rules are specific that Recuperating is required if the character ends a session still affected, so this whole discussion is only relevant in non-league games. Situations where Medicine can (in a home game) potentially be applied to a patient are out there, but are not something likely to happen more that every few sessions in a lot of campaigns.
The common factor seems to be that Constitution saving throws can eventually throw off effects of some diseases or poisons, or sometimes result in death with enough failed saves. The Recuperating activity gives Advantage on these throws, and should aid in recovering from those diseases or poisons that have a lingering effect. Medicine might conceivably be applied to any poison, disease, or injury that imposes an effect and provides repeated Constitution saves to end the effect.
In XGtE, pg 130, optional rules for Herbalism kit proficiency give advantage in using the Medicine skill proficiency to treat illness and injury (suggesting it's RAI to be able to do it), but is unclear how that is done with Medicine proficiency to start with. Number of rolls, frequency, the DC, the impact on the Con save, all are not defined. No help there in crafting a house rule.
All the diseases listed in the DMG seem like candidates for Medicine being potentially able to help.
Cackle Fever has saving throws to avoid damage, and a saving throw once a day to recover. Medicine might be employed to help mediate the damage and end the effect.
Sewer Plague has a daily Constitution save to recover. Failing adds exhaustion and the subject can't recover until they succeed on enough saves in a row to lower their exhaustion to one.
Sight Rot has an herbal cure, so that's one where Herbalism kit proficiency is more relevant, but it's possible the DM would allow a Medicine check instead.
These Bites cause diseases with lingering effects and repeating saving throws once a day, under RAW magical healing, potions, or Recuperating are probably required:
- Blue Slaad claws spread "Chaos Phage," a disease that takes several days to run its course, which ends in the victim transforming. It's clear that once transformed a wish spell is needed to reverse the transformation, but perhaps less clear if Medicine or even magic might cure the disease and save the victim from transforming. MM pg 276
- Red Slaad claws implant eggs/parasites considered a disease. The egg must be destroyed before symptoms present, after which the tadpole emerges within 24 hours. Magic would certainly resolve this, but given the time involved medicine is also at least plausible. MM pg 276
- A DC 20 Wisdom (Medicine) check can be used to remove a slaad control gem. MM pg 274
- A Gas Spore's Death Burst doesn't provide additional saves, but lasts for hours before killing the subject. There is potentially time for Medicine to help, but purely at DM discretion. MM pg 138
Keep in mind that a lot of poisons only give one chance to resist them, and Antitoxin is the best proactive option. The list of poisons where a medic might be able to help after the injury is shorter than you might think:
- Burnt Othur Fumes: immediate action needed, damage repeats each turn
- Carrion crawler mucus: the save to end the paralysis repeats each turn
- Pale Tincture: save repeats every 24 hours
- Ettercap bite: save repeats every turn, gives poisoned condition for one minute
- Deep gnome poisoned dart: same as ettercap bite
- Chuul tentacles: same as ettercap bite
- Bearded Devil Beard: same as ettercap bite
Other than Pale Tincture, these effects are too short term for Recuperating to come into play, but if stabilizing the dying takes a turn it doesn't seem impossible that Medical intervention could take the same amount of time.
The Lingering Injuries optional rules on DMG pg 272 includes Festering Wound which takes ten days and a series of DC 15 Wisdom (Medicine) checks to heal.
These wounds have an effect (usually called an Infernal Wound) that continues to do damage until a successful Wisdom (Medicine) check stops the bleeding:
If being able to non-magically "treat a patient" interests you, make that part of a session zero discussion with the DM to figure out the mechanics, or if it will even be allowed. You could look into how Heal checks worked in prior editions and discuss from there.
Medicine also has a lot of uses that don't really have a "patient" involved. Think of it as a specialized investigation-forensics skill used just like Nature to know or find things. That gives it value whether or not it can be used to end a disease or the poisoned condition. Personally I might pick it for the RP flavor on some characters, where not knowing some medical lore would be odd.