Limbs or body parts that are cursed off by Dark Magic cannot grow back. Dumbledore's hand had been destroyed by Voldemort's powerful dark curse placed on the ring Horcrux. It may be that that rendered his hand too damaged to receive a new or prosthetic one:
‘Will George be OK?’
All Lupin’s frustration with Harry seemed to drain away at the
question.
‘I think so, although there’s no chance of replacing his ear, not
when it’s been cursed off –’
and
Harry saw a clean, gaping hole where George’s ear
had been.
‘How is he?’
Mrs Weasley looked round and said, ‘I can’t make it grow
back, not when it’s been removed by Dark Magic. But it could
have been so much worse ... he’s alive.’
Deathly Hallows - pages 64 and 66 - Bloomsbury - chapter 5, The Fallen Warrior
Perhaps Dumbledore might have been able to be fitted for a prosthetic hand, but the exact parameters for having prosthetic devices in Potterverse isn't known. Also, JKR herself says not all things can be fixed by magic. I don't know if self-injury would be considered Dark Magic if magic happened to be involved in someone removing their own limb. You'll note that Wormtail cut off his own hand using a plain old knife and didn't use magic to sacrifice his hand. Perhaps that is the reason why Voldemort was able to give Wormtail a magical prosthetic hand -- Wormtail's hand wasn't cursed by Dark Magic while Dumbledore's was.