fI'd first like to note that the earliest games to be based around Star Wars - West End's d6 SW RPG and Decipher's Star Wars RPG, despite both having access to lots of background material, including Lucas's notes, never mention Force Healing as a thing. They're also relatively uncorrupted by later additions to the series (both predate the prequels, and West End's game even predates the expanded universe). They're both pretty scrupulous in maintaining accuracy to the movie universe, too. That neither of these sources ever references some sort of generic Force healing as a thing is pretty conclusive that no such power is referenced in either the movies themselves or the script or background notes. (And in fact, West End's SW sourcebooks are some of the best 80s source material for official background material because of the extensive access they got to otherwise unpublished documents at the time).
It's also not clear that Luke even needs healing in that scene in A New Hope. He's been knocked unconscious. There's no bleeding or other obvious trauma. (Which doesn't make a more serious injury impossible, just that we have no evidence one way or another).
A far better interpretation of the scene is that Obi-wan is checking to see if Luke's okay - not by searching for a pulse, but by using a Force power we know exists (sense) to check Luke's mental state. He determines Luke doesn't need medical care, and then shakes him a little to help him come to (after speaking to R2).
Possibly also noteworthy - the musical cues are not suggestive of healing Force powers at that point in the scene. The music has an edge of danger, emotionally pushing us towards wondering if Luke will be okay, and suggesting Ben is wondering the same. It's only as Ben tells R2 that he'll be fine that the music switches over to the Force/Obi-wan theme (it was obi-wan's theme in ANH, it only gets reified into the Force theme in later movies). If Obi-wan was doing something proactive with the Force at that point, you'd expect the musical cues to have shifted into Obi-wan's theme already.