What can a painting do? Try this. Here's the Rev Jeremiah Wright, in Trinity United Church of Christ, Chicago, 20 years ago. He's preaching about a picture. It's an image of a woman, who's sitting on the world, bent over, blindfolded, holding up a harp – "bruised and bloodied, dressed in tattered rags, the harp reduced to a single frayed string... and yet consider once again the painting before us. Hope! That harpist is looking upwards, a few faint notes floating upwards towards the heavens. She dares to hope. She has the audacity... to make music... and praise God... on the one string... she has left!"

The painting, as you may recognise from this slightly inaccurate description, is Hope by GF Watts. And the account comes from Barack Obama's memoir, Dreams From My Father, where this particular sermon is recalled as a turning point in the President-elect's life. It's not often that an art review can claim such a timely link. It's not often that an artwork can boast this kind of political effect. So maybe Watts himself will benefit from this testimonial from the most popular man on the planet. Nowadays, he's a painter who could do with all the help he can get.

George Frederic Watts, 1817-1904, is one of those unfathomable Victorian giants. He was "England's Michelangelo". He was an artist-prophet, whose subject can fairly be summarised as Life, the Universe and Everything. His motto was "The Utmost for the Highest".

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He was impossibly high-minded, unbelievably admired – and then, after his death, quickly a laughing stock and forgotten. Hope is the one that got away, the Watts image that has entered wide circulation. The rest of his work is in limbo. His pre-Raphaelite contemporaries have kept up a keen public following, but if you're interested in Watts, you're probably an art historian.

If you are, and in London, there's plenty to see. The National Portrait Gallery has his Hall of Fame, a series of whiskered portraits of fellow great Victorians, inspirational in purpose. Tate Britain has some of his most ambitious metaphysical images – She Shall Be Called Woman, The All-Pervading, The Dweller in the Innermost, a version of Hope – which it displays quite often. And in Kensington Gardens, there's the monstrous equestrian statue Physical Energy.

And now there's more. The Watts Gallery in Surrey is closed for improvements, and a big choice of its collection is being shown in the Guildhall Art Gallery and (from December) in St Paul's Cathedral. Another version of Hope is among them, and a broad view of his career is available. It looks like Watts's moment.

To grasp the audacity of Watts, though, and the absurdity of Watts, you need to be selective. Don't linger on his early career – sensitive portraits, rhetorical scenes from history. The important work begins around 1870, when, fired by Greek sculpture and Michelangelo, and Titian too, he embarked on a symbolic, mystical art that would awaken his viewers to the highest (a rather indefinable Watts watchword). He planned The House of Life, a mural sequence that would lay out his full world vision. His artistic allegiances were traditional, but his project was a distinctively megalomaniac modernist dream: art, the new religion.

Watts wanted to create a mythology. He drew on both Christian and Classical sources, and on the newly fashionable psychical research, but his goal was a universal super-mythology, transcending creeds, a pantheon of very general archetypes. Sometimes, as in Hope, there's a specific allegory to be decoded and moralised, but that's not where a picture stops. Watts sought the widest suggestiveness. He believed in a pictorial language that was non-literary and non-naturalistic. He said he was painting not reality, but sensations – or again, thoughts. He was dealing not with meanings but with forces, to be transmitted directly from the painting to the viewer's mind.

It is not quite abstraction. Watts never abandons imagery. But he pursued something that many modern abstract painters also pursued: an art that would convey states of mind and states of being, and operate subliminally, beneath the level of conscious response. His images are giving form to the motions of spirit. It's an art of elemental gestures and energies: bursting, rising, unfolding, enfolding, falling, being bent, twisted, crushed.

What's striking is how near, in some works, Watts gets to speaking this language pure. There's a painting from the 1870s called Chaos, or occasionally Cosmos. It's a panoramic landscape, peopled with titans that are half-blended into the earth, and the scene is in a process of convulsive unfolding from the formless to the formed. If it looks familiar, it's because in this scene Watts is inventing our now standard idea of the primeval terrain (though we'd have dinosaurs, not giants). But Chaos needs only a small retuning to become an imageless painting of the idea, the feeling, of striving evolution. It was to hang at the threshold of The House of Life.

It's the same way with the divine radiant sunbursts that dominate After the Deluge and Progress, or the general upward soaringness of She Shall Be Called Woman (in Tate Britain). The Sower of Systems, painted at the very end of his life, goes the furthest reach – a picture of some kind of striding creator-God, dissolving into a turbulent stirring of streaks of light and dark. But it's doubtful whether these pictures could completely dispense with their vague and minimal imagery. They need something to focus the visual gesture, to indicate that they're about matters of the highest importance.

But goodness knows, this is an art that needs to dispense with something. For the obvious thing about Watts's most ambitious work is that it's a catastrophe, ruined by self-inflicted confusions. Looking at these pictures, it's often possible to believe that he's the worst painter in the world.

Whatever he does ends up in a coagulated sticky murk and blur, in which the potential force of his grand gestures are bogged down and disabled – and worse, the murk and blur look like desperate evasion, a vagueness used to disguise the fact that Watts's painting of bodies is helplessly weak. Yet he will keep on doing anatomy, and doing it clearly enough to make its incompetence plain. He doesn't have the nerve to do full-on glimmering spooky gaseousness, except in a landscape like Sunset on the Alps.

And what can you say for a vision like Progress, with its rider advancing out of a sun? How did Watts manage to do the horse so that it looks like a horse's head attached to a duvet – didn't he want it to be too realistic? How did he bring himself to do a blazing solar disc and then destroy its colour-power with a smeared patina of brown all over – he thought it looked "old masterly"?

He's a fatally divided painter. He throws in everything, just in case. And it becomes a fascinating spectacle in itself, looking at this artist who wants to get through to something huge, to take art beyond what art can imaginably do, and just doesn't know how to do it.

As he knew. Watts's greatest work isn't a painting at all, it's a paragraph in which he faces up to this impossibility. "My attempts at giving utterance and form to my ideas are like the child's design, who being asked by his little sister to draw God, made a great number of circular scribbles, and putting his paper on a soft surface, struck his pencil through the centre, making a great void. This is utterly absurd as a picture, but there is a greater idea in it than in Michelangelo's old man with a white beard." Absurd in 1900, yes – but pointing again straight into the future of 20th-century art. The prophet prophesied.

GF Watts: Victorian Visionary, Guildhall Art Gallery, London EC2 (for information, follow Arts & Culture link from www.cityoflondon.gov.uk/corporation , or call 020-7332 3700), to 26 April; GF Watts: Parables in Paint, St Paul's Cathedral, London EC4 (01483 810 235; www.wattsgallery.org.uk/exhib_parablesinpaint.html ), 1 December to 30 July