All perfect praise be to Allah, the Lord of the worlds. I testify that there is none worthy of worship except Allah and that Muhammad, sallallaahu ‘alayhi wa sallam, is His slave and Messenger.
It is impermissible for the visitor of such statues to bow to them, even slightly. Bowing to these statues is plain imitation of idol-worshippers.
If a Muslim knows that a country compels its visitors to do this, it is impermissible for him to travel there in the first place. He should seek knowledge in any other country, since knowledge – whether it be related to history or otherwise – is not monopolized by these oppressive countries. It is impermissible for a Muslim to bow to anyone, even if done for mere greeting, and even if done without bowing the back. If it is disliked to bow to a human being, as stated by scholars, what then would the ruling be with respect to bowing to statues and idols?
An-Nawawi said, "Question: Bowing the head to a person... is it lawful or not? Answer: Bowing the head is disliked... ." [Al-Fataawa]
Hawaashi Shurooh Al-Minhaaj reads, "The custom of lowering the head and slightly bowing in a manner that does not amount to full bowing is not Kufr (disbelief), nor is it prohibited; however, it should be disliked."
Some scholars even stated that it is prohibited to bow to a person, so it is prohibited to bow to statues with greater reason.
The Permanent Committee for Iftaa' was asked, "We joined a karate club in America. The coach said, 'You have to bow when your opponent bows to you.' We refused and explained that this is against our religion. He agreed but said, 'You have to bow the head only, because the opponent initiates a greeting by bowing, and you must greet him back.' What is your opinion on this?"
The answer reads, "It is not permissible to bow to greet a Muslim or a non-Muslim, whether it is done by lowering the upper part of the body or the head, because bowing is a greeting of worship, and worship must be to Allah alone."
Allah knows best.