This has often been alleged, but doesn't have much of a basis in actual evidence from the medieval period. While no one appreciated being shot in the eye with an arrow or a bolt, they took missile weapons very seriously and with a great deal of respect. The position of Master of the Crossbows in France was a prestigious royal title granted by the king himself. Although the Master would not be a crossbowman himself, no one would dare say to a royal officer's face that his position was in any way dishonorable. In Flemish towns and cities, shooting guilds were an important source of crossbowmen for local armies. Their members would have been drawn from the ranks of the burghers, guild craftsmen and skilled tradesmen. Their upper leadership would be even wealthier and people of civic importance within the town or city. Far from disrespecting these crossbowmen for their choice of weapon, the Dukes of Burgundy made every effort to demonstrate their respect for the shooting guilds and sometimes even participated in their activities such as shooting contests. The Dukes of Burgundy valued their crossbow militias not only for their practical military value, but also because their civic importance meant that maintaining alliances with them lessened the chance that the cities would rebel against outside authority. If word got around that the Duke or his men were demeaning the honor of the guilds, tensions between the Duke and the cities might rise.

In England, of course, the late medieval kings loved their bowmen because they were a cheap and effective way of raising troops quickly. A guard of 24 archers formed the bodyguard of the king, who were often granted access to his person and served as royal officials on various projects. Many other members of the royal household (tradesmen like fletchers, armorers, tailors, etc.) would also be archers who served the king both in peace and war. This force, along with the knights and men-at-arms who were part of the king's personal retinue, would form the nucleus of any royal English army of the period. Other aristocrats would also have their own retinues of soldiers, many of whom would be archers connected to their households. Recruitment in the English army during the late medieval was done through local networks of friends and family connected to the military community. A captain attempting to recruit a company for his lord (or forming a free company on his own) wouldn't want to risk insulting potential recruits, who might decide to take their skills elsewhere and bring their relatives along with them. Professional archers were cheaper than hiring men-at-arms, but still a significant expense. They were hired all across Europe as mercenaries by many different polities. If they were disrespected by anyone, it would more likely be because they were professional soldiers who murdered and looted everything around them rather than because they were archers.

I've hear this trope again and again, and I don't know about any of the other medievalists but I have never come across anything leading me to believe medieval cavalry had a disdain for ranged weapons, unless of course they got pinned down by them.

Much what has been transmitted to us of the "Knightly Code of Honor" or whatever you want to call it was a twelfth century concept primarily developed in France during the reign of Philip II, who was arguably the architect of the modern French state. During Philip's reign the French Monarchy took large steps towards asserting itself over the nobility, strung together a series of defeats against the Angevins, and granted several rights and privileges to the bourgeoisie. Given these conditions, it is unsurprising that the nobility, especially the petty nobility, sought to reaffirm their status by proclaiming that "true" knights behaved in a certain way that made them superior, based on an idealized concept of the behavior of Charlemagne and his paladins five hundred years before.

We could say a lot about what Chivalry was and wasn't, but the point is that as far as I know it did not include disdain for ranged weapons. The connection is easy to make given its other characteristics, but combined arms tactics were simply too useful to be ignored. In Italy especially, even in the reign of very francophile Dukes of Milan like Galeazzo Visconti or his son Gian Galeazzo, combined arms were a central component of military doctrine: armed citizenry were very often deployed armed with crossbows. In the Republic of Venice too, which had a little-talked about but very real chivalric and knightly culture, the armed citizenry was primarily armed with crossbows (however this has more to do with the the fact that most Venetian military engagements were naval prior to the 15th century). And longbows and crossbows, respectively, were important in the military doctrine of England and France: we all know the story of the battle of Agincourt.

Both MI13 and AlviseFalier make a fair assessment of the medieval world and the use of range weapons. However this question isn't about the military theory of the time or the values of general medieval society, but about the personal opinion of knights. It really remined me of a passage in Jean de Joinville Chronicles of the Crusade. He was a participant of the battle of Al Mansurah (1250) during Louis IX Seventh Crusade and wrote down his experience. He describes the battle quite realistic, he notices the fear and death but he closes with a passage that changes the tone of the description quite a bit:

(Jean de Joinville along with some of the other knights had sped in front of the main cavalry force pursuing the enemy but had found themselves outnumbered and had been fighting for their lives. Their horses had died at this point. Just at that moment the main force arrives.)

As I stood there on foot with my knights, wounded as I have told you, King Louis came up at the head of his battalions, with a great sound of shouting, trumpets, and kettledrums. He halted with his troops on a raised causeway. Never have I seen a finer or more handsome knight! He seemed to tower head and shoulders above all his people; on his head was a gilded helmet, and a sword of German steel was in his hand. The moment he stopped, those good knights in his division whom I have already named to you, together with other valiant knights of his, flung themselves right at the Turks. It was, I can assure you, a truly noble passage of arms, for no one there drew either bow or crossbow; it was a battle of maces against swords between the Turks and our people, with both sides inextricably entangled."

I remember this passage mainly because he normally isn't that positive about Louis. But it also answers your question in a way. Jean the Joinville judges this battle more noble then any battle he fought, even though he lost it. The reason for that is the absence of archers and crossbowman. (The food troops were stuck on the other side of the river Nile and did not participate in the battle.)

So while Jean de Joinville doesn't denounce archers as useless or cowards in any way, he certainly doesn't think their way of bearing arms is as deserving of respect as that of him and his knights. That being said, while I havent checked, Iam fairly certain that besides the knights in his employ Jean de Joinville also employed an amount of missile troupes on this campaign. He might idolise the knightly war, but is smart enough to know the value of archers and crossbowman.

I've heard this was true for the armies of the ancient Greek city-states, that ranged skirmishers were disdained and dishonorable, but necessary in their armies. Can someone with the relevant expertise confirm or deny that? If it was true, it's possible that the author is drawing on inspiration from a bit further afield.

We might as well hammer this nail in so far that we'll never have to do so again. (heh)

It's my understanding that for the knight, there was a time and place for all types weapons. Like a carpenters toolbox, you mainly use the hammer and knife, but all tools have their place. Let's look at some texts which talks about archery and crossbows for nobles in the north.

In a 13th century didactic text to the prince, we're told countless times about the bow, in training and different application for it's use.

It is also counted rare sport and pastime to take one's bow and go with other men to practice archery.

Indicating that the Prince, and his friends, already owned a bow. We're also given account of effective weapons on shipboard:

Weapons of many sorts may be used to advantage on shipboard, which one has no occasion to use on land, except in a fortress or castle. Longhandled scythes and long-shafted broadaxes, "war-beams "and staff slings, darts,: and missiles of every sort are serviceable on ships. Crossbows and longbows are useful as well as all other forms of shooting weapons; but coal and sulphur are, however, the most effective munitions of all that I have named. Caltrops cast in lead and good halberds (ed: atgeir, a weapon often associated with throwing) are also effective weapons on shipboard.

We're also described ranged combat from mount:

there are, however, other weapons which a mounted warrior may use, if he wishes; among these are the composite bow and the weaker crossbow, which a man can easily draw even when on horseback, and certain other weapons, too, if he should want them.

While all this isn't descriptive, that is, it's not talking about events which took place, but it's from a book which aspires towards a broader courtly culture of the continent. It's goal is to lend credence to the other royal courts, how to behave, etc.

Also, in the courtly lawcode which address' the king's liegemen, we're again reminded that the knights are advised to own a bow or crossbow. The lesser nobles are required to bring a bow.

We're also told, from Richardson's Phd Thesis that English nobles were gifted fine crossbows.

Most of these crossbows were clearly of munition type, but the detailed account of the manufacture of fine crossbows for gift by the king to members of the nobility during the 1340s suggests they may have been used by men-at-arms during sieges.

We can see here that ranged weapons are not something which is taken in disdain, but is part of a wider arsenal of weapons that the knight may use when the context is right. They're not praised beyond other weapons, nor are they downgraded, but in a special context they are part of your kit. The educational text mentioned above ends the discussion of weapons training with this fitting sentence:

Formerly the custom was for all who wished to become expert in such arts and thoroughly proficient in war and chivalry to train both hands alike to the use of weapons. Strive after the same skill, if you find yourself gifted for it, inasmuch as those who are trained in that way are the most perfect in these activities and the most dangerous to their enemies.