There have been many reports of the Philosopher’s Stone over the centuries, but the only Stone currently in existence belongs to Mr Nicolas Flamel, the noted alchemist and opera-lover.
Philosopher's Stone - page 161 - UK Hardcover - chapter 13, Nicholas Flamel
The stone didn't belong to Dumbledore. It wasn't his prerogative to destroy it alone. I think by having it placed in the Gringotts vault, Nicolas Flamel agreed to have the stone protected (presumably from Voldemort) by Dumbledore.
At the end of Philosopher's Stone Dumbledore and Nicolas Flamel decided together that destroying the stone was for the best.
‘Oh, you know about Nicolas?’ said Dumbledore, sounding quite delighted. ‘You did do the thing properly, didn’t you? Well, Nicolas and I have had a little chat and agreed it’s all for the best.’
Philosopher's Stone - page 215 - UK Hardcover - chapter 17, The Man With Two Faces