Synopsis: Nope; other than speaking the language of snakes, he did no magic. And speaking the snake's tongue may not even count.
He did use Parseltongue to open the door, but I'm not sure if that counts as using magic.
Harry approached, his throat very dry. There was no need to pretend these stone snakes were real; their eyes looked strangely alive.
He could guess what he had to do. He cleared his throat, and the emerald eyes seemed to flicker.
“Open,” said Harry, in a low, faint hiss.
The serpents parted as the wall cracked open, the halves slid smoothly out of sight, and Harry, shaking from head to foot, walked inside.
After that, he never had a chance to cast spells; the incarnation of Tom Riddle picked up Harry's wand right after he came thru the door, as Harry knelt to look at Ginny. Tom had the wand for the entire fight, only relinquishing it when he was vanquished.
Riddle didn’t move. Harry, sweating, managed to hoist Ginny half off the floor, and bent to pick up his wand again.
But his wand had gone.
“Did you see — ?”
He looked up. Riddle was still watching him — twirling Harry’s wand between his long fingers.
Then, after the battle...
There was a long, dreadful, piercing scream. Ink spurted out of the diary in torrents, streaming over Harry’s hands, flooding the floor. Riddle was writhing and twisting, screaming and flailing and then —
He had gone. Harry’s wand fell to the floor with a clatter and there was silence. Silence except for the steady drip drip of ink still oozing from the diary. The basilisk venom had burned a sizzling hole right through it.
So, yeah.. He could have done the whole thing as a squib, as long as he could speak 'Open' in parseltongue... Which we saw later could be done by a non-parseltongue, as Ron did exactly that in Hallows; he just had to have heard it before to know what sounds to make.