We will focus here only on the Shabbat-related issues, about which you ask.[1] Relatively recent poskim have discussed this matter, and the main discussions concern the halachot of medicinal activity on Shabbat and the melacha of tzovei’ah (coloring).
The issues regarding one who wants to sunbathe due to a specific medical condition have too many variables to address in this forum. If the sunbathing is for (perceived) general health benefits, the matter is similar to the halachic discussion about vitamins on Shabbat. The Shulchan Aruch[2] rules that one may eat/drink medically-oriented foods that are eaten regularly by healthy people. However, the Magen Avraham[3] contends that this is so only when one is eating the food for its food value, not when he is doing so specifically for its medicinal value. The Igrot Moshe[4] maintains that this stringency applies only when the person, while not sick, needs strengthening, but not when he is simply trying to keep his body “well stocked” so that he will not weaken in the future. Some poskim are even more stringent than the Igrot Moshe.[5] In any case, since medicinal activity is forbidden on Shabbat only when it is evident that the health benefit is the objective,[6] and since sunbathing is something that people in our times do not usually associate with health benefits, it is permitted, at least on this count.[7]
The question of coloring the skin is fascinating. The prohibition of coloring is relevant to the human body, as we infer from the fact that certain cosmetics are forbidden on these grounds.[8] Despite the fact that it would certainly not be forbidden to go outside to a place where one could get a suntan, perhaps it is forbidden to purposely get one, especially if one takes steps to increase the extent of the coloring. Indeed, some poskim forbid intentionally getting a suntan.[9] The Shemirat Shabbat K’Hilchata,[10] however, implies that it is permitted because the person does not color his skin actively, but rather puts himself in a place where the sun does the work. Others add that the process is drawn out and the act itself has no immediate impact.[11] (Some melachot, such as cooking, always involve putting an object in a place where an outside force will have a gradual impact on it. Coloring, in contrast, is classically done by putting the coloring agent directly on the object; the question is whether coloring by tanning – a passive, gradual process – is also problematic.) Another claim is that tanning causes a natural, not artificial, color for the skin.[12]
It is hard to find conclusive proof from classical sources on these claims. However, from our perspective, a general principle of “halachic philosophy” tips the scale. Since it is unreasonable to assume that the Torah and Chazal forbade walking in the sun on Shabbat, it is difficult to imagine that the Torah/Chazal extended the issur of tzovei’ah to include coloring the skin in the sun, which often happens as a result of being outside, regardless of whether one likes tanning. Consequently, even intentional tanning would be permitted.
Therefore, while we are not particularly enthusiastic about the prospect of someone spending a good part of his Shabbat sunbathing, we would not forbid it on halachic grounds.
Do realize that according to the consensus of poskim, it is forbidden to smear sunscreen cream over one’s body or part of it, due to the melacha of memachek.[13] [14] One cannot justify using sunscreen on the grounds of the medical importance of protecting one’s skin from the sun if the need stems from one’s desire to sunbathe (in other words, simply don’t sunbathe). It is permitted to spray liquid sunscreen.[15] This is not considered medicinal because its purpose does not involve healing.[16]