No wonder Roy Lichtenstein announced his breakthrough in a 1961 painting called Popeye. As the spinach-fuelled sailor fells his opponent with a mighty-muscled fist, we realise that the artist was aiming a similar blow at the sensibilities of his viewers. After the sublime abstractions of Jackson Pollock and Mark Rothko, Lichtenstein's brazen fascination with cheap cartoons and commercial images seemed unfor-givably debased. He was condemned as a purveyor of trash, and yet pop art swiftly confounded its detractors by ballooning into the most talked-about movement of the mid-Sixties.

Before attaining this bold maturity at the age of 40, Lichtenstein had been painting in an oddly hesitant, Francophile manner for more than a decade. But the Hayward Gallery's retrospective starts, wisely, with the moment when he cast away doubts. In another, equally feisty painting from 1961, a triumphant Donald Duck brandishes his fishing rod and exclaims: "Look Mickey, I've hooked a big one!!" The same words could have been uttered by Lichtenstein, who surely shared Mickey's grin as he worked on this big, brash canvas.

He took a subversive delight in choosing banal subjects, culled at first from advertisements and sales brochures. Nothing could have been more ordinary than a golf ball, yet Lichtenstein blew it up into a ruthlessly simplified painting. Isolated in emptiness, the ball resembles a much-dented planet floating in the cosmos. Like many of his initial pop paintings, it was executed in sober black and white. Another monochrome work called Radio, complete with leather strap and aluminium stripping, juts out from the wall like a relief sculpture.

Lichtenstein went on to explore three-dimensional forms later in his career, and I regret their absence from this show. But back in 1962, he succumbed to the lure of comic-book romance. Taking one frame from each love story, he turned their corny theatricality into images of mystery. We have no idea why the blue-haired, flinty-jawed young man swivels away from the aghast blonde behind him, growling: "Forget it! Forget me! I'm fed up with your kind!" Lichtenstein forces us to speculate.

Some of the romance pictures centre on a ridiculously glamorised version of the art world. Another blonde, this time coy and simpering, turns to "Brad darling" and predicts that his new "masterpiece" will "have all of New York clamoring for your work!". We smile at her gushing enthusiasm, yet the painting uncannily anticipates Lichtenstein's own success. There is nothing self-indulgent about the rigour of his pictorial organisation. He displays an absolute command of essential contours, and the Benday printing technique used in comics gives him a ready-made means of peppering his canvases with hailstorms of dots. For all their outrageousness, these paintings are formidably disciplined.

One fascinating display case at the Hayward contains some of the comic books Lichtenstein worked from, including the frame he used as a springboard for his immense and cataclysmic air-battle scene Whaam!. The source for this painting was a book bearing the unironic title All-American Men of War; and from Lichtenstein's pencil study displayed alongside, we discover just how much he chose to exclude and refine. Although he retained one of the victorious pilot's speech bubbles, and faithfully copied its gung-ho words, Lichtenstein deleted the other one - perhaps because its pseudo-poetic comment sounded bogus: "The enemy has become a flaming star!" But he enlarged the explosion drawn so tentatively in the comic book, allowing the flames to surge apocalyptically. He also intensified the diagonal thrust of the attacking jet, as well as cancelling the word "Whoosh!" from underneath the plane. Lichtenstein must have thought that "Whoosh!" would distract our attention from the word "Whaam!", which he duly enlarged and painted in blazing yellow.

Studying the comic books also makes us appreciate how cleverly Lichtenstein zeroed in on the essence of a scene. An arresting picture from 1964 concentrates, in powerful close-up, on a young red-haired woman clutching her phone as she mutters: "Ohhh . . . Alright . . ." We are left to wonder why she looks so tense and crestfallen, whereas the original, showing a far less stunning woman, reveals exactly what she is hearing through the receiver: "I'm sorry, Nancy, but I'll have to break our date! I have an important business appointment." The words sound dull enough to suggest that Nancy is lucky to be liberated from her humourless boyfriend, but in Lichtenstein's painting the redhead's anguish might well have been triggered by news of a traumatic event.

The emotional range of these women is surprising. The fur-coated blonde passenger who speeds past us, in a streamlined painting called In the Car, looks haughty and remote from her handsome driver. But the girl in Cold Shoulder goes much further, lowering her eyelids as she utters a frosty "Hello" in a speech bubble dripping with five white icicles. Even her pearl necklace looks glacial. By contrast, the wind-blown blonde in a spectacular painting from 1965 flushes with pink Benday dots as she hurtles through the city thinking: "M-maybe he became ill and couldn't leave the studio!" Yet even she remains impeccably stylish as she holds up an elegant gloved hand to prevent her golden tresses from becoming too ruffled.

As the crisply installed Hayward show proceeds, all this metropolitan melodrama drops away. We are confronted instead by vastly enlarged splashy brush marks, suspended on a field of black dots so that the vitality of each luscious stroke becomes at once monumental and momentous. Lichtenstein's imagination was saturated in the work of other artists, and he loved playing games with iconic images from the past. Monet's Rouen Cathedral series is subjected to a blizzard of dots, and in the central canvas the great medieval facade almost disappears in a fiercely optical onslaught. Filled with a sense of imminent dissolution, it compares well with Lichtenstein's later large and over-elaborate paintings such as Cosmology or Figures in Landscape, where he cluttered the picture surface with an embarrassment of stylistic quotations.

By the 1990s, Lichtenstein had recoiled from this excess. He recovered his old simplicity in a sequence of teasing yet tranquil interiors, and then, only a year before his unexpected death, he embarked on a surprising series inspired by oriental art. They are fugitive images of mountainous landscapes, viewed as if through veils of mist. Although the dots proliferate and almost take over, they are still infinitely adaptable and make subtle references to the natural world. At the base of several paintings, Lichtenstein inserted a clearly defined boatman, bridge or tree. But they remain diminutive and threatened by the engulfing haze around them.

All the exclamatory action in his early pop pictures has vanished now. It is as if Lichtenstein arrived at a supremely meditative, dreamlike way of seeing just before he succumbed, with shocking suddenness, to his final illness. In this respect, the late oriental scenes are eerily prophetic, leaving us in the end with serene and stoical intimations of the void.

"Roy Lichtenstein" is at the Hayward Gallery, South Bank, London SE1 (0870 169 1000) until 16 May