I believe that chivalry is inextricably linked to Christianity because the Church was such a part of the formative process of chivalry that it would not have been chivalry without it.
After the collapse of the Carolingian Empire in 888, France, Germany, and Italy fragmented into a feudal mess. Local lords and knights struggled to maintain their demesnes while vying for power over their peers. Internecine feuding resulted in raids on each other, with the peasantry and the clergy feeling the brunt of these attacks. As a result, in 989 the Synod of Charroux issued the Pax Dei or Peace of God, under the authority of Archbishop Gombald of Bordeaux and Gascony, with agreement from the bishops of Poitiers, Limoges, Perigueux, Saintes, and Angouleme. It declared the nobility, including the knights, were to refrain from robbery of the poor and defenseless (the peasantry) of their provender and livestock, robbing or assaulting the godly (the Clergy) who were not bearing arms, and the plundering of Church property, upon pain of Excommunication. To this were added assaulting or robbing women (especially virgins and widows) and children, burning houses, beating the peasantry, and robbing merchants. The Pax Dei spread into all of Western France, including Aquitaine, Burgundy, and Languedoc by the early eleventh Century, and was strongly supported by King Robert II of France.
While the effectiveness of the Pax Dei was limited at its inception, it led to the Church issuing various decrees, including the Treuga Dei (Truce of God), which prohibited fighting by knights and nobility on Sundays, Holy (Feast) Days, all of Advent, Lent, certain other days of the week (Thursday, in memory of the Ascension; Friday, in memory of the Passion; and Saturday, in memory of the Resurrection), and the days between April 25 to Ascension Thursday, and the eight days leading up to and including Pentecost. Also, that all church personnel and property, pilgrims, women, merchants and their servants, and peasants working in the fields and their livestock were under permanent peace.
During the eleventh century, the Church pressured the nobility to accept trappings of piety regarding knights. Building off of St. Augustine of Hippo's City of God, where the idea of a just and godly war to defend God, if no other means of conflict resolution was available, was just and right; Odo of Cluny wrote the Life of St. Gerald of Aurillac, which described the perfect "knight of Christ". Chansons de geste, such as the Song of Roland, showed how noble knights lived their lives not only as fierce warriors but paragons of spiritual virtue. Knights were encouraged to turn over spoils of war to the local church or monastery as offerings in thanks of God's divine protection on the battlefield. The second son of a noble family was often destined for service in the Church, strengthening the bond between the nobility and the clergy. Squires (in the sense of knights in training) were to hold vigil, praying and fasting the night before his accolade, his armor and weapons on the altar to be dedicated in service to God. Homage oaths were sworn with God as witness, as breaking said oaths would invite divine retribution. Sword blades, crosses, and pommels were engraved with Scripture or dedications to various saints. Many knights upheld the Virgin Mary as a paragon of virtue and dedicated themselves to her, which influenced courtly love and honorable treatment of women. God was always watching, providing protection for those who were virtuous and providing retribution against those who were evil.
Even with all this, the nobility were often flirting with disaster for their souls. Finally, Pope Urban II declared a Crusade to the Holy Land, to wrest Jerusalem from "the grasp of the infidel". Tens of thousands took the Cross and made their way to Outremer at the close of the eleventh century. It not only solved Europe's problem with excess bored knights with nothing better to do than to fight each other by sending them to the Holy Land with promises of wealth and righteousness in the eyes of God, it also cemented the Church's influence on the knighthood. With the victory of the Crusade and the establishment of the kingdom of Jerusalem under the Cross of Christ, that influence was now adamant.
As chivalry evolved during the twelfth century, certain people were deemed by medieval Christian authors as exemplars of knightly conduct: Godfrey of Bouillon, Sir William Marshal, Bertrand du Guesclin, even Saladin, who was deemed chivalrous, regardless of his religion. The development of military orders in the Crusades, themselves founded by the Church, influenced chivalry during this time. Arthurian legend changed in the hands of Chretien de Troyes and others to the medieval romance, where the characters of Gawain, Percival, Galahad, and Lancelot as definitions of chivalry became popular. Chivalry eventually came to be accepted as duty to God, King, Liege, fellow Christians, and women.
While one can argue that codes of conduct can be established without religious influence, it would be difficult to prove that what we know of as chivalry would have come about without the efforts of the Church. Compared to other codes of conduct, such as bushido, the code of chivalry maintains an air of virtuousness that still inspires today, even in a society where such ideals are often considered passe.