Whether you believe in the spiritual power of religious relics or not, the British Museum's exhibition of Catholic antiquities contains some wonderful art, says Adrian Hamilton

To hold an exhibition of sacred Roman Catholic relics in a country with quite so secular and Protestant a history as Britain's was always going to be a delicate affair.

The reliquaries themselves are glorious enough, many of them masterpieces of craftsmanship. Relics were big business in the Middle Ages. Armed troops were sent to seize them, emperors vied with each other to amass great collections, and the churches that claimed them drew huge crowds of pilgrims to worship them.

Which was the problem, of course, for the Protestants. As Martin Luther wryly observed of the remains of Saint Barbara's skull: "If anyone counts up the pieces, she will have seven heads." John Calvin went further, writing a whole treatise against relics, asking: "How do we know we are venerating the ring and comb of the Virgin rather than the baubles of a harlot?"

The answer to that is one of faith. If you believe you are viewing a real saint, as 100,000 people did in queuing over four days for a sight of Saint Therese of Lisieux's relics at Westminster Cathedral only two years ago, then there is something profoundly moving in this act of veneration. If you don't believe, then these are empty and even ridiculous expressions of ancient beliefs.

Which poses the question for any museum: do you present them as art objects, things of beauty and fascination in their own right, or do you display them as spiritual vessels helping us to understand and feel what the thousands who once pressed to view them believed?

It's a quandary that was acknowledged from the beginning of this magnificent exhibition's journey, which started in Cleveland and Baltimore in the US, and is now ending in London. Is it, the organisers kept being asked, going to be too religious? "I think it's fair to say that in America the display veered towards the artistic side," says the British Museum's curator, James Robinson. "Here we're trying to be more balanced in both directions – art and devotion."

That may owe something to the personality of the British Museum's director, Neil MacGregor, a Roman Catholic with a long-held interest in religious art. It was he who held the Seeing Salvation exhibition at the National Gallery to mark the millennium, and it was its extraordinary success last winter (nearly 360,000 people went to see it over 10 weeks) that helped the National Gallery do what I thought no British gallery would ever do: hold an exhibition of full-on Spanish Catholic statuary in its The Sacred Made Real exhibition.

It was right to try. Over a century of progressive secularisation, we have lost much of the understanding of not just the beliefs but the spirit behind objects of veneration. The great altarpieces of the Renaissance have become separated from the altar itself, and what it and they represent. This especially applies to reliquaries, for virtually all religions have the veneration of saints and relic pilgrimage as a tenet of their faith. Yet go to any exhibition of Buddhist, Jainist or Islamic art, let alone Christian objects, and it is as if what they represented and still represent in spiritual terms is now devoid of meaning. It is art for art's sake.

The art in this exhibition, is very good. As you ascend the stairs to the show in the old Reading Room, you encounter the near-lifesize Reliquary Bust of St Baudime from Saint-Nectaire in the Auvergne. Staring straight ahead with an impassive gaze, he raises his arms in welcome and benediction.

Gold was the symbol of purity and incorruptibility, and each jewel had a special meaning, which is one explanation for the opulence of so many of the reliquaries that so appalled the Protestant reformers. The other is the belief that items of great spiritual worth deserved the best craftsmanship and materials, and the relics business gave churches the money to do so.

Right across what is now Europe, monasteries, churches and cathedrals laid claim to "authentic" pieces of Saints' bodies, and fragments of the cross, the crown of thorns, and the robes and the chalice associated with Christ and the Virgin Mary. For the most credulous, simply to touch or look upon them might cure them of disease, provide children and ensure good health. For the others, to the spiritual embracing of relics brought them closer to Christ.

Britain, of course, was ruthless in hunting down and casting out such idolatory. Determined to make religion an adjunct to temporal power, Henry VIII's agents sought to itemise and destroy reliquaries. But across most of the rest of Europe, even Scandinavia, reliquaries survived in surprising numbers.

The organisers of this exhibition have selected objects from a number of different places to demonstrate the various devotions and styles. They don't duck the more uncomfortable facets of relic devotion – the arms with bones encased in crystal, the fragments of a body encased in jewels. Indeed, there is a whole section devoted to "Speaking Reliquaries". But most of these works were created not so much to glorify the objects as the holiness surrounding them. There are somefabulous shrines in Limoges enamel, and copper gilted, showing the influence of Byzantium on Western European art well into the Middle Ages, as well as works from Byzantium itself.

As the Middle Ages progressed, so the work became more self-consciously artistic. A couple of South Netherlandish busts from the early 16th century have a sweetness that's very different from the painting of the period, obsessed as it was with the suffering and realism of Christ's crucifixion. Most striking of all is Reliquary with the Man of Sorrows from mid-14th century Bohemia, the figure of Jesus, sorrowful but sublime.

Veneration of relics didn't die out with the Reformation, although later it morphed into a worship of more secular "saints" – such as Lenin, Marilyn Monroe and Che Guevara. The exhibition ends with a short film making this point. It's somewhat specious. From a modern perspective, you could equate the veneration of saints with the contemporary "worship" of celebrities but that is to ignore the presence of God and his incarnation. In the end, the British Museum shies away from this. It has captions about the saints and spaces devoted to the relics but it doesn't capture the sense of fervour with which mediaeval pilgrims shuffled round countless churches to commune with God through the remains of martyrs and saints.

Instead, it has opted to categorise this as part of a series on spiritual journeys starting with the Egyptian Book of the Dead exhibition earlier this year, followed by a show about the Muslim Hajj – religion pigeonholed as a cultural phenomenon. No disgrace in that. State museums are secular institutions in a secular age. We may not relate to the religious spirit in this show but we can view a collection of wonderful objects.

Treasures of Heaven, British Museum, London WC1 (020 7323 8181) to 9 October