The French King when following the étiquette of the court in Versailles would have a very public life. But it was ritualized and not every aspect of his life was public, or would have been with the entire court (some ceremonies were restricted to a fairly restricted audience). As the duke of Saint-Simon (possibly the most famous memorialist of the late reign of Louis XIV), said : *With a calendar and a watch you could tell from 300 miles away what he [the King] was doing". It should be noted that this system was not invented by Louis XIV but that it developped progressively from the early XVIth century to reach its final complexity in the Louis XIV era.
The whole point of this system was that by granting access to the King (and the possibility to have a very quick talk to him) and by allowing the King to show his favour for this or that courtier, it was possible for the King to organize the court in a complex and detailed hierarchy of préséance (i.e. who has precedence in dignity and social standing over whom). Every small privilege (for instance being authorized to attend the King's grand lever) was a sign of recognition sought after. This was an essential part of the domestication of French nobility in the absolutist monarchy after the Fronde that achieved its final form under Louis XIV. As the Grand Louis himself wrote in his Memoirs : « Les peuples sur qui nous régnons, ne pouvant pénétrer le fond des choses, règlent d'ordinaire leur jugement sur ce qu'ils voient au dehors, et c'est le plus souvent sur les préséances et les rangs qu'ils mesurent leur respect et leur obéissance ». (The people on which We reign, as they cannot grasp the full extent of things, usually base their judgment on the most exterior aspects of things, and most of the time, it is based on préséances and rank that they measure their respect and obedience). The overarching importance that questions of préséance would have had for a noble living in the court of Versailles is well exemplified in the mémoires du duc de Saint-Simon who spends many pages describing the petty warfare of it, with who had stool sitting privilege in specific conditions, and his personal opinion on the matter. He was famously very opinionated on such questions^1 (and deeply resented that honors were granted to the bastard children of the King, and the faction he considered defending them), and also spent many pages spewing biles on the bastard Maine (one of the legitimized sons of the King), the pro-bastard Dubois (a non-noble cardinal and political figure of the Régence), and the pro-Dubois Noailles (who despite noble ascendance had sympathies for the former). A good example also in Saint-Simon's work is the affaire du Bonnet (where there was debate as to whether the First President of the Parlement de Paris had to take off his hat or not when saluting a duke and pair de France). It went to court and after long proceedings gave reason to the President, which infuriated Saint-Simon.
To summarize such moments in a day of Louis XIV (the étiquette would be somewhat toned down under Louis XV and XVI but its core structure and the system of préséance would survive until the Revolution) :
The King would awake at 8am : for the first 15mn, he is only surrounded by the first valet, 6 chambergrooms, the first surgeons, and the premier gentilhomme de la Chambre du Roi (a nobleman) who would be allowed to pull the bed curtains.
Then at 8.15am, there would be the petit Lever : first could enter (by the back door) the princes du sang, the first doctor, the first surgeon, the first valet. Then could enter (by the main door) officers of the Crown, the grand Chamberlain, the Grandmaster of the Wardrobe and the First Valet of the Wardrobe (all noblemen) an a few chosen courtiers the King wanted to show his favour. At this moment at least 22 persons were in the room, by that count. Then followed a very quick Catholic prayer (15mn) which the King would follow from his bed. The came the barber and the whig cabinet Valet : the King then chose a whig and a pair of slippers, put on his bedrobe ; the Chamberlain could then take off his sleeping bonnet, and he would be shaved and brushed.
Then followed the Grand Lever at 8.30 : came in the ordinary surgeon and doctor, the first valet and the comptroller and warden of Silverware, and the first valet of the Wardrobe. The King would seat on his "business chair" (chaise d'affaires, literally) to do his business (i.e. take a shit). The courtiers that had been granted a brevet d'affaires (a specific visa allowing them to meet the King on this occasion) were then introduced to discuss with him. Then came in the Grand Aumônier, Ministers, Councillors of State, Marshals of France, the Grandmaster of Game (Grand Veneur), the Grand Louvetier, the Grandmaster of Ceremonies. Then came the people of quality (gens de qualité). At that step, at least 50 person were in the room vying for the Monarch's attention.
Meals were taken with only very few people seated (petit Couvert would have had only him and his brother seated, Grand couvert would have had only the direct royal family).
The coucher would follow essentially the same ritual, in reverse (with less people somewhat), and with the Bougeoir ceremony (everynight, a different noble was allowed to hold and extinguish the candle for the King, which was a very sought for honor).
Regarding sexual activity, the King would enjoy some privacy for affairs ; the only relatively public business he conducted in that matter would have been consummation of marriage, where, while there would not have been any courtiers in the room, the question of whether it took place was something of significant interest as non-consummation was a cause for annulation (and then, major diplomatic crisis). In general, as the King and the Queen had separate apartments, any nightly visit would have been witnessed (and quickly known to most of the Court).
^1 : Le point de vue de Saint-Simon sur les rangs, Jean Duquesne, Cahiers Saint-Simon Année 1999 27 pp. 71-78
^2 : A. Grellet-Dumazeau, L'affaire du bonnet et les Mémoires de Saint-Simon