To kill an animal is said to be a sin in the Sanatana Dharma.
So, why did Shree Ram shoot the arrow at the deer when asked by Sita mata? What is the justification for this action?
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Why did Rāma shoot the arrow at the deer when (simply) asked (to be captured) by Sītā?
Because instructions from Sītā were to either capture the deer alive or hunt it for its golden hide (if it cannot be captured).
Chapter 3 (Sarga 41)
When he [Lakshmana] spoke in this way to Kakutstha, Sita, the one with the beautiful smiles, restrained him. Her senses confused by the deceit, she cheerfully said, 'O son of a noble person! This delightful deer has stolen my heart. O mighty- armed one! Bring him here. We will play with him. ... O king! But I have not seen a deer like this earlier. ... This extraordinary and wonderful deer is stealing my heart.
If this deer allows you to capture him alive, that will be extraordinary and will generate great wonder. When our residence in the forest is over and we return to the royal palace again, this deer will be an ornament in the inner quarters. O lord! This divine form of the deer will generate great wonder in Bharata, in the son of the noble one and in my mothers-in-law.
O tiger among men! If you are unable to capture this excellent deer alive, its hide will still be beautiful. Even if the spirited one is killed, I wish to be humbly seated on its golden skin, spread with tender darbha grass. It has been said that this kind of ferocity and conduct driven by desire is unseemly in women. But the form of the spirited one has generated great wonder in me. His body hair is golden. His horns are adorned with the best of jewels. His complexion is like that of the rising sun and is as radiant as the path of the nakshatras.'
On hearing Sītā's words, Rāma is also overcome by greed:
Raghava's mind was also filled with wonder. He heard Sita's words and saw the extraordinary deer. Happily, Raghava addressed his brother, Lakshmana, in these words. 'O Lakshmana! Behold what has caused desire and delight in Vaidehi. This is the best among deer and his like does not exist today in the forest, in the region of Nandana, or in the refuge of Chaitraratha. O Soumitri! Where on earth is there a deer that is his equal? The beautiful hair on his body grows up at some places and down in others. With colourful spots of gold, this deer is splendid. ... Can you name anyone whose mind will not be tempted by this deer? His form has the complexion of molten gold. There are many kinds of celestial jewels. On seeing him, whose mind will not be filled with wonder? On hunts, kings roam around in the great forest. O Lakshmana! Wielding bows, they kill deer for the sake of their flesh. They exert themselves in the great forest and collect many kinds of minerals, gems, jewels and gold. All those riches make men prosper. O Lakshmana! Everything thought of in the mind enhances Shukra's treasure house. O Lakshmana! If a person desires artha and goes around unhesitatingly collecting that artha, those who know the sacred texts about artha say that this artha is true artha. With me, the slender-waisted Vaidehi will be seated on half of this deer's golden and gem-encrusted hide. ... There are only two deer that are so beautiful and divine. There is this one that is wandering around on earth. There is also the divine one that roams around in the sky, following the path of the stars.
Also, he thinks that if the golden deer happens to be the māyā of Mārīca, the rākṣasa, then going after the deer gives him an opportunity to eliminate Mārīca who has been troubling the sages in the forests. He says:
O Lakshmana! Even if what you have said is correct, that it is the rakshasa's maya, it is my task to kill him. This is the cruel Maricha who is unclean in his soul. He used to wander around in the forest earlier, causing violence to the bulls among the sages. There were many kings, supreme archers, who came here on hunts and he arose against them and killed them. Therefore, this deer should be slain.
Remain here. Arm yourself and attentively protect Maithilee. O descendant of the Raghu lineage! Everything that we wish to do is based on her. I will kill or capture this deer. O Soumitri! I will go and quickly bring the deer back. O Lakshmana! Look at Vaidehi and see how she desires the deerskin. The deer's skin is his remarkable feature and he will no longer remain today. Without any distraction in sentiments, remain in the hermitage with Sita. I will slay the spotted deer with a single arrow. O Lakshmana! I will kill him and quickly come back with the hide.
What is the justification for this action?
If Rāma was simply going after Mārīca because he thought Mārīca needed to be eliminated to establish dharma, then his actions are justified. He even made a promise to protect the sages of the Daṇḍaka forest earlier in Araṇya-kāṇḍa.
But to go after the deer for its golden skin when he is supposed to live like a sage during his exile, seems unjustified. His reasoning – kings roam around in the great forest ... they collect many kinds of minerals, gems, jewels and gold ... all those riches make men prosper – maybe valid in general, but he was not an ordinary king during his exile.
You may also be interested in this answer where I defend Rāma's actions.
This is described in detail in Valmiki Ramayana: Aranya Kanda: Sarga 43.
Sita asked to capture the deer alive as it was so beautiful that she wanted to take it to Ayodhya after their vanavas.
But, Laxmana suspected that this deer must be some Rakshasa and Rama then said that if this happened to be Rakshasa, it will be killed.
And, Rama didn't shot the arrow as soon as he spotted the deer. After a long discussion, he appointed Laxmana to protect Sita and went to search the deer.