And now Harry stood in the headmaster’s office yet again. It was nighttime, and Dumbledore sagged sideways in the thronelike chair behind the desk, apparently semiconscious. His right hand dangled over the side, blackened and burned. Snape was muttering incantations, pointing his wand at the wrist of the hand, while with his left hand he tipped a goblet full of thick golden potion down Dumbledore’s throat. After a moment or two, Dumbledore’s eyelids fluttered and opened. “Why,” said Snape, without preamble, “why did you put on that ring? It carries a curse, surely you realized that. Why even touch it?” Marvolo Gaunt’s ring lay on the desk before Dumbledore. It was cracked; the sword of Gryffindor lay beside it.
-- HP and the Deathly Hallows, chapter 33: The Prince's Tale
It was putting on the ring, not destroying it, that affected Dumbledore's hand so badly. Destroying it using the sword of Gryffindor, in itself, had no ill effects. So there was no reason to believe destroying the locket would curse Ron in the same way Dumbledore had been cursed.
That said, Harry was aware that the piece of soul inside the locket might attempt to 'fight back' in some way, as the piece inside the diary did (by trying to kill him in Chamber of Secrets) and the piece inside the ring also did (by tricking Dumbledore into putting the ring on, surely). He says so explicitly to Ron:
“I’m going to open it,” said Harry, “and you stab it. Straight away, OK? Because whatever’s in there will put up a fight. The bit of Riddle in the diary tried to kill me.”
-- HP and the Deathly Hallows, chapter 19: The Silver Doe
And indeed, the piece inside the locket does try to fight back, by taunting Ron and creating physical visions to infuriate him. Fortunately, Ron is strong enough to withstand its temptations and destroy the Horcrux as required.