The BA Medical kit mentioned earlier (formerly, and still known to crew as the M5 kit) has an indemnity form within it. BA will indemnify any medical professional who acts on board their aircraft.. can't speak for other airlines, which obviously doesn't mean to imply no-one else does. I think you will find amongst the larger carriers this is the norm. If I'd thought, I could have looked for the form, as had the kit open last trip (buccastem).. will make a point next time.
Also several major airlines have contracts with medical advice agencies (Medlink in Phoenix being the one I am familiar with.) Contact can be made by Satellite phone if fitted, HF radio, or ACARS (SMS-type datalink.) Have used this service in anger twice and found it invaluable where there was no on the spot medical personnel. Crew are trained to note basic signs and symptoms, and guided by the use of a pro-forma; information is then passed to the Doctors who are aware of aircraft position, route and destination as well as the nature of any specialist hospital units along the intended route.
Indeed, the value can come not only in advising of the best diversion options, and urgency of the situation, but also in sometimes preventing an unnescessary diversion where we as flight crew may have been inclined to err too conservatively based on lack of relevant information.