God said to be stewards of the animals (Genesis 1:28), which means taking care of animals and not killing them. Euthanasia would therefore go against God’s instructions. It ends an animal’s life prematurely. We should pray for miraculous intervention (James 5:15) and do our part to heal and care for animals until God takes them rather than ending the life when it seems right in our own eyes that it should end (Proverbs 3:5-6, Judges 17:6).
Euthanasia tends to take on two primary forms. One is ending a terminally ill or incurably injured animal. The other is ending a shelter animal’s life that no one is willing to adopt and care for. The former is an act that stems from a desire to end an animal’s suffering. The latter is a desperate last resort when no one is willing or able to take on the responsibility of being the abandoned animal’s steward.
Therefore since the people euthanizing animals are, generally speaking, acting from a place of compassion or desperation rather than from a place of cruelty, I see euthanasia as a much less pressing issue than the rampant and unapologetic cruelty to animals that is so normalized in society and even within the church (Habakkuk 2:17). So although euthanasia goes against God’s ideal of stewardship and we are better off abstaining from doing it, the issue of intentional animal cruelty is much more pressing because it is indicative of a heart that lacks compassion (Proverbs 12:10) whereas euthanasia is not.