This is "Quest" by Lee Harding. It was originally published in New Worlds Science Fiction, April 1963. You can read it at the Internet Archive. If you read it in an anthology, it may have been Lambda I and Other Stories (1964).
The protagonist, Harry Johnston, starts out the story talking to the Divisional Controller:
"Something real," he said. "Something that hasn't been made by man. Something that isn't synthetic. That's all. Not to keep. Only to see. So that I will at least know that it is there. Where can I find such a thing?"
He starts by travelling around, but always comes back to where he starts; then he decides to go down, only to discover there's nothing not manufactured down there either.
"He had nursed the tiny hope that perhaps in the depths of the world he might discover rock and earth and soil in their natural condition. But there was nothing. Only the ever-present product of man's industrial genius. And behind the wall thrummed the energy of the mighty machines that made possible the existence of the miles of city overhead.
He turned around, defeated. "I think I'll go back."
"Very good, sir."
A sudden though occured to Johnston. "How far down are we?"
He repeated the figure to himself. "Is this the lowest level?"
"If you mean, does the city extend below us, no, it does not, sir."
Mister Johnson stopped and tapped the floor with the toe of one shoe.
"Then what's down there?
"Several miles of insulatory material."
"And after that."
"An archaic term that describes the inner core of the planet. That is all. There is...nothing more."
He starts to trek northward on foot for several days until he collapses, and is convinced to take an air-car. The towers of the city gradually start to reduce in height and buildings spread out, but only briefly until another massive conurbation starts. He climbs to the maximum height and after hours he suddenly notices a spot of green.
Abruptly, the Great Park exploded across his eyes. He recoiled from the assault of greenery that filled his world, and punched the descent button eagerly.
He eagerly explores the park, looking at the trees and shrubs, marveling at the ants in the grass and the bird flying overhead. He finds a garden of flowers and meets the Caretaker. As he leaves the Caretaker's cottage he has the impulse to pluck a rose growing nearby:
By the doorway was a tremendous rose bush. Scarlet blooms burst hungrily towards the sunlight. A sudden desire swept over him and he stretched out a hand to pluck one of the flowers to carry with him, next to his heart.
"No." The cry from the Caretaker shattered the solitude abruptly.
But the flowers aren't real:
Defiantly he curled his fingers carefully around a thorn-free stem of a rose and plucked it quickly from the bush. He held it up in plain view of the Caretake and sniffed the delicate perfume arrogantly.
In his hand the rose withered and died. Dead leaves crumpled into a wraith-like, gossamer remnant of the texture of a spider's web.
He despairs and cuts his veins to kill himself, but he's not actually alive either:
His veins were empty, already collapsing.
And still he lived. The pulsating awareness buried within his skull had no need of the external carapace designed only to delude his conscience and his fellow men.