Who makes the final decision on wether or not Candy's old dog should be shot ?

Ultimately, Slim's reaction, or lack or reaction makes the decision to shoot the dog a finality.

Candy looked a long time at Slim to try to find some reversal. And Slim gave him none. At last Candy said softly and hopelessly, "Awright—take ‘im." He did not look down at the dog at all.