I don’t think this is stated explicitly in the books, but there are fairly strong hints that Dumbledore didn’t try to use the Stone again.
When he explains the Hallows to Harry in the “limbo” at the end of the final book, he talks about how he was “unworthy” to use the Stone. This is the passage:
After another short pause Harry said, “You tried to use the Resurrection Stone.”
Dumbledore nodded.
“When I discovered it, after all those years, buried in the abandoned home of the Gaunts — the Hallow I had craved most of all, though in my youth I had wanted it for very different reasons — I lost my head, Harry. I quite forgot that it was now a Horcrux, that the ring was sure to carry a curse. I picked it up, and I put it on, and for a second I imagined that I was about to see Ariana, and my mother, and my father, and to tell them how very, very sorry I was…”
“I was such a fool, Harry. After all those years I had learned nothing. I was unworthy to unite the Deathly Hallows, I had proved it time and again, and here was final proof.”
— Deathly Hallows, chapter 35 (King's Cross)
This seems to be genuine contrition and regret on Dumbledore’s part. Unlike many people, Dumbledore seems to set store by one’s “worthiness” to use the Hallows, and if he really recognised that he was unworthy to use it, then he would not do so.
Then later, he talks about using the Stone as a potential action, rather than something he actually did:
“Maybe a man in a million could unite the Hallows, Harry. I was fit only to possess the meanest of them, the least extraordinary. […] The stone I would have used in an attempt to drag back those who are at peace, rather than to enable my self-sacrifice, as you did. You are the worthy possessor of the Hallows.”
— Deathly Hallows, chapter 35 (King's Cross)
Both of these suggest to me that he didn’t try to use the Stone again.
As you point out, the Stone worked normally after it was detached from the ring, so I think it was self-control that kept him from using it, rather than any magical defence.