He was born Edwin Parker Twombly Jr in Lexington, Virginia, in 1928 - Cy was his father's nickname. A self-portrait photograph taken as a young man, with a long exposure that imitates the rich, slow-maturing tones of a Victorian photograph, has him looking out of time: an archetypal southern exquisite, his left arm draped languidly over a chair, his shirt collar not quite neat. He might be a character out of a southern gothic fiction, hypersensitive, haunted, creating the most tender art and poetry while listening for dead fingers scratching at the walls of the mausoleum in the swamp.
Time, love, doomed desire and a prolonged unravelling are the themes of the art of Cy Twombly. "Ah, it goes, is lost," he has scrawled in pencil on one of the four tall canvases that constitute his Quattro Stagioni (Four Seasons), A Painting in Four Parts (1993-94), now on view at Tate Modern. This is an artist who stands aside from the boisterous, rowdy, gung-ho assertion of American art in his lifetime, an artist who seems to inhabit a different universe, another country - as indeed he does, having settled in Rome in 1957, just as American art attained its empire.
Twombly's art is fervently, tenderly, even comically romantic, elegiac, pleasurably mournful. It relishes gaudy, dangerously exposed emotional language, with words written loosely on the canvas - graffiti, but of the most literate and gentlemanly kind, steeped in the classics. In The Four Seasons, a great and moving modern masterpiece whose purchase was made possible by the American Fund for the Tate Gallery and which should become one of Tate Modern's treasures, this language has a heady analogue or correlative or perhaps merely arbitrary accompaniment in crocuses, wounds and rainstorms of sensuous colour: bloody spring, yellow-white summer, pink and olive autumn, yellow and pine winter blasted by dreadful white. The word "happiness", used too many times, becomes hysterical, even menacing, as it is repeated beside the red vessels - depictions of ancient Egyptian boats - on his panel Spring.
Time passing is the inescapable burden of any series of paintings entitled The Four Seasons. In the history of art (with which Twombly is obsessed), images of the changing times of the year, the countryside in different seasons, suggest an optimistically cyclical view of life. What dies is reborn, as spring will follow winter, when everything makes love and the world, pulsing with life, floats in the fruit-studded forest of Botticelli's Primavera.
And yet all of Twombly's Four Seasons - though distinct and tasting of the different times of year - share a mood of decline, loss, consciousness of that which must pass. In high summer, the artist stares into whiteness, a memory of love dissolving in the mad heat. If anything, the best time of the year, Twombly's evident favourite, is autumn, when the wine of memory can be savoured, the mellow fruits of previous action enjoyed; desire is sweetest in its echo. Autunno is the most relaxed of these canvases; there is the purple blush of wine, the vintage of the year in soft flowers of colour. Twombly has said that the wine harvest in Bassano in Teverina, a village north of Rome, inspired this panel, and that this was the beginning of his idea for the series, which he thinks of as starting with autumn.
At Tate Modern, the four canvases are hung more conventionally - Spring, Summer, Autumn, Winter. Spring is where we should look first for the larger meaning of this ambitious "painting in four parts", a strange idea that stresses the fragmentary and the incomplete. Twombly's Primavera belongs not to the warm, lovely new beginnings ordained by heaven that medieval and renaissance art celebrate, the season of Chaucer's April with its sweet showers and Botticelli's Primavera, but to modernism's travesties of a natural sense of time: TS Eliot's parody of Chaucer, "April is the cruellest month, breeding/Lilacs out of the dead land", and Stravinsky's blood-sacrifice ballet, The Rite of Spring. Twombly's Spring is a beginning and a birth, and, most of all, an erotic awakening. There is red-hot sex in this painting, as Egyptian-style boats studded with spear-like, jagged, red oars flame out of the whiteness. The words in pencil beside them feel like some overexcited, ecstatic love letter.
Yet Spring is also violent: those boats are war canoes. They resemble the ships in Twombly's recent cycle of battle paintings, Lepanto, and invoke history, recalling not just north American war canoes but "solar" boats placed in Egyptian tombs. Twombly feeds on the past, history, archaeology, myth. Like Turner, whose melting, body-shaking light Twombly cites in his Summer canvas, he uses the sense of time conveyed by the moods of nature to suggest historical seasons and inevitable transformations: the rise and fall of empires, the making and destruction of human power, the wars of love.
Two years ago Twombly awed the Venice Biennale with Lepanto, and currently he stands with Richard Serra - whose massive steel walls couldn't at first sight be more different from Twombly's work - as one of the two most productive and indispensable senior American artists. Both artists matter now because they make monumental, historical art at a time of pervasive amnesia.
The difference between them is that, while Serra works from within the tough, aggressively serious tradition of American art that began in the 1940s with Jackson Pollock, Twombly is at a tangent to that tradition: he is the least American of great American artists. He is as remote from what American art is supposed to be as Gainsborough is from Damien Hirst. American art has always had a Yankee puritanism in it - think of the strident severity of Barnett Newman, the wrath of Mark Rothko, the deathly gaze of Andy Warhol. There is something against this grain in Twombly's painting, with its fetid eroticism, ostentatious quotation of European art, myth and history, and perhaps most of all its dandyish historical pessimism. As America started to see itself as the new world power, Twombly chose to leave, to linger amid the ruins of ancient Rome. And the sense of history as a cycle in which power will always collapse is fundamental to his art: history as erection, expenditure and fall.
Of course, you can find clues to the strangeness of Twombly's art in what little he has allowed to become public of his life. An intense friendship with his fellow southerner, Robert Rauschenberg, was a dominant influence on his early art (and Rauschenberg's). With Rauschenberg he attended Black Mountain College in North Carolina, the legendary artists' seminary where figures such as John Cage, Robert Motherwell and the poet Charles Olson confronted the new American art of the 1940s and 1950s with the ideas of the European avant-garde, especially Duchamp. And it was with Rauschenberg that Twombly first travelled to Italy, Morocco and the Mediterranean world his imagination so loves. Twombly shares with Rauschenberg a propensity for coded autobiography and a taste for the impure, the complicated - and they both have a poetic sense of history. But where Rauschenberg turned to collage, assemblage, an architecture of dense quotation, Twombly allowed only one impurity into his paintings: writing.
Writing is implied in the marks and scrawls of abstract expressionist painting - Pollock even called one of his early pictures Stenographic Figure - but it took Twombly to replace the spontaneous mark with the written word. His words are passionate, but as language they are also received: he did not invent the word "happiness". So the declared spontaneity and unauthorised, savage originality of American art is replaced, quite casually, in Twombly with an insistence on the historical nature of art, without for one second becoming sedate.
And yet, when you recite the art-historical contexts and the nuggets of suggestive biography, Twombly remains an enigma. In the army in the 1950s he worked on cryptography, later saying he was too "vague" to be much good at it. Perhaps that is a warning against becoming too analytical and exegetical about his work. Twombly's paintings are florid, romantic and ornately overstated; they are baroque, decadent concoctions; they are the paintings Edgar Allan Poe's Roderick Usher might have done; they are sensuous flowers of the south; they are seriously unedifying, poisonous vintages. They are letters to history from the heart of a corrupt empire.
· Cy Twombly's Quattro Stagioni is at Tate Modern, London SE1 (020-7887 8008).