Could they? Yes. Would they? Not likely.

A key facet of medieval European political thought was the idea that the king (or, yes, occasionally a queen regnant) was God's representative on earth, and within their kingdom was supposed to embody certain virtues or characteristics. Related was the concept of the "great chain of being," by which all of the world was organized into a strict hierarchical structure, with God at the top, followed, by the angels, followed by humans, followed by animals and so forth. Each category had its "primate," or superior being -- among the angelic orders the leaders were the seraphim, and within the worlds of mankind the leaders were the crowned heads of Europe.

So for most kings (at least, the smart ones who weren't inclined on getting excommunicated, which was serious trouble), it was good to at least be outwardly good. Still, in spite of all that Neoplatonist mumbo-jumbo, medieval elites had a very close relationship with prostitution and sex work in general. Why? Well, a lot of people realized that having a population of young, unmarried men become sexually frustrated was probably a bad idea, and in order to head off any trouble they would turn a blind eye to prostitution, because the alternatives (fornicating with married women, homosexuality, and so forth) were considered to be much worse.

Augustine of Hippo once apparently remarked of the sex trade that "if you expel prostitutes from society, you will unsettle everything on account of lusts." I have been unable to find the treatise in which he said those specific words, but that comment has been widely quoted, particularly in the book Sex, Dissidence and Damnation: Minority Groups in the Middle Ages by Jeffrey Richards which does a pretty good overview of prostitution (chapter 6). A whole bunch of kings took this to heart and either tolerated prostitution or even opened their own brothels, a trend seen across England and France. The prostitutes of Southwark were known as the "Winchester Geese" because their patron was the Bishop of Winchester! (see Richards but also this Smithsonian article on the history of sex work in Southwark). True, there were kings that sought to either restrict or outlaw prostitution altogether -- King Henry VI of England tried to purge Southwark of its prostitutes in 1460, sending its less-than-reputable residents to prison, while King Louis IX of France (you know, Saint Louis) issued a number of decrees before going off on crusade in 1269. Of course, this didn't work, and most kings and city authorities ended up throwing their hands in the air and allowing a particular street or neighborhood, preferably outside of city limits, to be a center for prostitution. In the case of the Spanish monarchs Ferdinand and Isabella, loyal supporters were granted rights to exploit sexual labor from various regions as a reward for service, as if it were any other profitable industry. (See Eukene Lacarra Lanz, Marriage and Sexuality in Medieval and Early Modern Iberia.)

Okay, we've established that if you walk through the average medieval European city you could probably find a bunch of prostitutes, unless there was a super big plague or the king went on a bit of a sanctifying spree very very recently. How many kings used them? Not many, in the historical accounts, although charges of sexual immorality were common in this regard. Sure, many kings kept lovers and favorites, male and female. Prostitutes, though? We have an incident from the life of Cesare Borgia, the son of Pope Alexander VI and scion of the notably dissolute Borgia family, who was accused of organizing what was called the "Banquet of Chestnuts." Basically, he hired a bunch of prostitutes (about fifty) to entertain his guests at a banquet. This story is attested from a single source and is probably apocryphal, but has been repeated many times because the Borgias did a lot of non-prostitution stuff that made people think "yeah, this sounds like something that they'd do." I don't think I can find any other instance in which a member of a royal or nearly-royal family (such as the Borgias) was documented as visiting prostitutes, much less a ruling monarch!

Robert Baratheon is pretty unusual in his sexual habits when comparing him to his counterparts in European history. Most kings were reluctant to run the risk of venereal disease, even before syphilis crossed the Atlantic and regardless of the fact that germ theory was centuries in the future -- everybody knows that sexual misconduct was grounds for getting cursed by God, after all. Maybe King Robert has it easier because his kingdom's religious authorities are more subservient or less inclined to condemn their monarch for openly visiting brothels. Still, his siring of many illegitimate children is highly irresponsible and quite accurately cited as a sign of his misrule; both in the ASOIAF universe and in our own history, royal bastards tended to cause considerable trouble in the succession of the realm. But that's another story.