It seems balanced, but some details could make it more consistent with existing ones.
It is a very versatile and broad homebrew, which could make it too strong. I'll address some details in the sequence, but personally I would remove the +1 Wis/Str bonus or allow the healing effect only once per rest (Note: the one that needs to rest is the one being healed, not you). There is a wall of text ahead comparing to some other feats in order to explain my reasoning.
First, let us compare the proposed effects with ASI. Then with other (existing official) feats.
For a class that depends on Str or Wis, if they are at an odd stat score, it is strictly better than ASI (ok, except for carrying capacity or some extremely edge cases), since it increases the same +1 to the modifier, gives +1 damage and allows healing.
If they are in an even score stat, the +1 Str/Wis does nothing for now. Then, let's see the trade-off for each score:
If you would get +2 Str, you get +1 Str instead, not getting the +1 Modifier. The main difference here is the +1 to attack roll that the PC won't get from your feat. Is the ability to free heal worth it? Probably. Specially since the Healer feat exists (will be addressed later).
If you would get +2 Wis, you get +1 Wis instead, not getting the +1 Modifier. The main difference here is for Spellcasting, which won't increase the Spell Save DC. You also don't get the +1 bonus for Perception, which might be huge depending on the party and encounters. But again, is the free heal worth it? Probably.
Compared to ASI, I would say your feat is better almost every time.
Other feats that give Half ASI
Most feats that give Half-ASI (i.e. +1 on ability score) and actually have some combat significance were added in XGE as Racial Feats. Most of them either are passive abilities that trigger when you do something (cast fire spells for the tiefling one, for example) or can be used once per rest.
Check Orcish Fury, which gives you +1 Str or Con.
When you hit with an attack made with a simple or martial weapon, you can roll one of the weapon’s damage dice an additional time and add it as extra damage of the weapon’s damage type. Once you use this ability, you can’t use it again until you finish a short or long rest.
As an action, you can spend one use of a healer’s kit to tend to a creature and restore 1d6 + 4 hit points to it, plus additional hit points equal to the creature’s maximum number of Hit Dice. The creature can’t regain hit points from this feat again until it finishes a short or long rest.
So, it's a 1d6 instead of 1d4 and the constant bonus (+4 + max number of HD) is always higher than your +Proficiency. Note my emphasis, though. Only being able to use it once for a creature until the next rest is a lot more restrictive than capping it at half HP.
On top of that, you also give +1 on Wis or Str and +1 on Damage Rolls. Compared to healer, your feat is considerably better.
As I mentioned in the start, I would either
- Remove the +1 Str/Wis, making it more consistent with the Healer feat - trading the extra burst healing (1d6 + 4 + HD) for a smaller but more frequent heal (1d4 + Prof) and +1 Damage.
- Allow it to be used only once per rest (of the healed creature), again making it more consistent with the Healer feat and other Half-ASI feats.
The reason I call it "balanced, but would change some details" instead of straight-forward "not balanced" is because the Healer feat is not a strong feat, so it being better than that doesn't say much. Same for being better than the Half-ASI feats from PHB - they are not exactly the best feats to start with. If you compare it to Sharpshooter, Sentinel or similar, it is not that strong.