Per the title: Could Fawkes's tears have healed Tom Riddle's diary after Harry stabbed it with a Basilisk fang?
As it was pointed out to me in Why Didn't the Basilisk Bite in Book Two Destroy the Horcrux, when Harry stabbed the diary Horcrux with the Basilisk fang, the diary wasn't destroyed instantaneously; there could have been time for Fawkes to cry onto the stab "wound".
[...] without thinking, without considering, as though he had meant to do it all along, Harry seized the Basilisk fang on the floor next to him and plunged it straight into the heart of the book.
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.
Chamber of Secrets - page 237 - British Hardcover - chapter 17, The Heir of Slytherin
So, would it have been possible for Fawkes's tears to have healed the mortally wounded diary Horcrux?