The following is from the script at this link.
It would seem that they had ridden Little Blackie beyond a point of restoration, even choosing to cut the horse with a knife to drive it on. Given the foaming, and the unsettling noises coming from within the horse, Rooster chose to end the horse's misery. The horse was nowhere near a place it could heal. In Rooster's mind, he most likely thought it wouldn't even if they were close.
Mattie is woozy. As Little Blackie crosses the field at
full gallop Mattie looks blearily at the littering bodies of horses
and men. Next to Lucky Ned’s body his horse, saddled and riderless,
swings its head to watch as Rooster and Mattie pass. Mattie’s eyes are
Mattie’s eyes half-open. Little Blackie plunges on,
through a rough road in woods, but slower now, his mouth foaming.
Rooster "Come on, you!" Mattie "We must stop." Little Blackie is played
out. Horrible noises are indeed coming from the horse, but Rooster is
grim: Rooster "We have miles yet." He leaves off whipping the horse and
takes out his knife. He leans back and slashes at the horse’s
whithers. Little Blackie surges. Mattie screams. Mattie "No!" A
locked-down shot as horse and riders enter at a gallop and recede.
It has started to snow. Mattie is flushed and soaked with
sweat. The horse is laboring for breath. Rooster gives inarticulate
curses as he kicks it on. Mattie looks ahead: Barely visible in the
moonlight a man mounted bareback rides on ahead. A sash cord holds a
rifle to his back. He recedes, outpacing us, disappearing into the
darkness and the falling snow. Mattie "He is getting away." Rooster "Who
is getting away?" Mattie "Chaney." Rooster "Hold on, sis." Mattie is
falling. It is unclear why. Her legs squeeze the horses flanks. Her
hand tightens on the horses mane. Rooster’s arm reaches around to hold
her. Little Blackie is giving out, going to his knees and then all the
way down. Rooster hangs on to Mattie as the horse sinks. He pulls her
clear, lays her on the ground, and then steps away from her, taking
out a gun. The horrible noises coming from the horse end with a
gunshot. Rooster reenters to pick up Mattie but she screams at him and
claws at his face, opening fresh gashes. He ducks his head as best he
can to avoid the claws but that is the extent of his reaction.