At Genesis 1:28, God told the first humans that all of the animals of the earth would be in subjection to them:
“Have in subjection the fish of the sea and the flying creatures of the heavens and every living creature that is moving upon the earth.”
God also made it clear that animals could be properly used and killed by man. For instance, their skin could be used as garments. God also gave permission for humans to eat the meat of animals after the Flood of Noah’s day, supplementing the diet of vegetation they were originally given. (Genesis 3:21; 4:4; 9:3)
Animal cruelty, on the other hand, is against God's will. Consider Nimrod at Genesis 10:9. The Bible describes Nimrod as “a mighty hunter”, suggesting the wanton killing of animals for sport. The same verse says that this put him “in opposition to Jehovah.”
Also, the Bible tells us that:
"The righteous one is caring for the soul of his domestic animal." (Proverbs 12:10)
However, this does not mean that God views animals in the same way that he does humans. The Bible indicates that God gave humans the hope of everlasting life, but not for animals. (See "Is there a hope of an afterlife for animals?")
The decision to not let a pet suffer needlessly because of very old age, severe injury, or terminal illness is up to a Christian's own personal decision. If he decides that it would be merciful not to let a pet continue to suffer without any reasonable hope for recovery, then he may choose to have it put to sleep.
For more, see:
Pets - Keep a Balanced View of Them
Search For Bible Truths - ARCHIVE
Scriptures Index
Search For Bible Truths - Search Guide