Question about the contrast between Buddhist statements and Christian doctrine

I think that Jesus's most famous saying, along those lines, was:

I am the way and the truth and the life. No one comes to the Father except through me.

I'm wondering whether the Buddha might make an analogous statement.

I think not because, as you said, the Buddha realized and taught emptiness or egolessness. Buddhist doctrine is that ego or views of "self" are a source of suffering (maybe some examples of egoistic views could include "I live", "I will die", "I am wealthy", "You have stolen from me", etc.).

The Buddha teaches that forms, and feelings and ideas, must be regarded as "This is not mine, this I am not, this is not my self."

O monks, the well-instructed noble disciple, seeing thus, gets wearied of form, gets wearied of feeling, gets wearied of perception, gets wearied of mental formations, gets wearied of consciousness. Being wearied he becomes passion-free. In his freedom from passion, he is emancipated. Being emancipated, there is the knowledge that he is emancipated.

There are several suttas where the Buddha refuses to be identified as anything in particular, except perhaps as Tathāgata, and a few other epiphets such as trackless.

Another difference is that Buddhist are expected to reach liberation by their own effort. There is a way towards that liberation: and what's special about the Buddha is that he discovered (or rediscovered) that path, and, he was able to teach it (some people discover it for themselves but don't teach it) ... that's not the same as saying though that the Buddha himself is the way.

There is some identification of the teacher (the Buddha) with the teaching, for example the following which the Buddha said to his disciples as he was passing away:

Now the Blessed One spoke to the Venerable Ananda, saying: "It may be, Ananda, that to some among you the thought will come: 'Ended is the word of the Master; we have a Master no longer.' But it should not, Ananda, be so considered. For that which I have proclaimed and made known as the Dhamma and the Discipline, that shall be your Master when I am gone.

But I don't think you see in that the same kind of "I am" message that you quoted in the OP.

To go a bit further, "I am" is a statement that's attributed to God ("I Am that I Am"). I think there's some similar "I Am" messaging in the Hindu doctrines of Atman and Brahman ... which, I contrast with the Buddhist "no-self" doctrine.

The word 'Buddha' means 'Enlightened'. It is the Buddha that is the light of the world. Buddhism states there can be only one Original Buddha in a world-system (MN 115). Jesus was a new light for Judah but not for the world. Jesus brought light to the violent tribal hypocritical religion of Judaism but Buddha (obviously; probably) brought light to Jesus.

When Jesus came to Judah and taught new teachings, which were different to Judaism, such as: (i) forgiveness, (ii) non-violence, (iii) showing the other cheek, (iv) non-hatred, (v) universal love & mercy, (vi) deeds before faith; (vii) God is love, (viii) celibacy, (ix) wandering like a monk, (x) not thirsting again (xi) liberation from the world, (xii) a kingdom not of this world, (xiii) eternal life (not experiencing death), etc, these new teachings obviously came from Buddhism because they have no source in the pre-Babylonian-exile Old Testament. All of these teachings have direct parallels in Buddhism, including the idea that 'God (Brahma) is love', which the Buddha taught to Brahmans in the Tevijja Sutta.

The Buddha revolutionized the ancient spiritual world, such as the famous Hindu book called the Bhagavad Gita (300BC), which is about Krishna, who is very similar to Jesus, being a human incarnation of god. Although the Bhagavad Gita refers to god & its theistic, its more profound principles are very similar to Buddhism (such as giving up craving & attachment).

The Buddha did not 'give up his life' for 'sheep' because the Buddha taught: (i) to not sacrifice one's life for another (Dhp 166); and (ii) he was the teacher of gods & humans (rather than animals or 'sheep'). Unlike Jesus, the Buddha forbade the practise of miracles for the religious conversion of others (DN 11). Unlike Jesus, a Buddha must live the whole term of his natural life as a demonstration of his virtue, i.e., his social harmony.

Unlike Jesus, the Buddha did not wander around picking fights with the religious establishment. The fact that Jesus (supposedly) violently died on the cross & was (supposedly) killed by Jews resulted in a religion that perpetuated violence and engaged in revenge killing against Jews for 2,000 years, which ultimately influenced the 'Holocaust'. This cycle of revenge was not one-sided, with (wealthy) Jews also plotting for hundreds of years to destroy Christianity. In short, the (supposed) sacrifice of Jesus has not really brought peace to the world but the opposite. Where as practitioners of the Buddha's noble path are peaceful & don't engage in conflict with other religions.

The Buddha called himself the 'Spiritual Friend' or 'Noble Friend' because he showed people a path for individual freedom that was self-reliant. The Buddha said:

Admirable friendship, admirable companionship, admirable camaraderie is actually the whole of the holy life. When a monk has admirable people as friends, companions, & comrades, he can be expected to develop & pursue the noble eightfold path. It is in dependence on me as an admirable friend that beings subject to birth have gained release from birth, that beings subject to aging have gained release from aging, that beings subject to death have gained release from death, that beings subject to sorrow, lamentation, pain, distress & despair have gained release from sorrow, lamentation, pain, distress, & despair.

Upaddha Sutta

In his first sermon & many other places, the Buddha described his teachings as 'light':

'This is the Noble Truth of the Cessation of Suffering': such was the vision, the knowledge, the wisdom, the science, the light that arose in me concerning things not heard before.

Dhammacakkappavattana Sutta

The following phrase is found many times in the Buddhist scriptures about the 'lamp' of the Buddha:

It is, good Gotama, as if someone were to set up something that had been knocked down, or to reveal what had been hidden, or to point out the right path to a man who had got lost, or to bring an oil lamp into a dark place so that those with eyes could see what was there — just so has the good Gotama shown me the truth in various ways!

And this, as Jesus taught (from Buddha):

Open are the doors to the Deathless to those with ears.

MN 26

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As for the term 'emptiness', it means 'empty of self', such as found in the term 'kenosis' in Philippians 2:7:

Who, existing in the form of God, did not consider equality with God something to cling to, but emptied Himself, taking the form of a servant, being made in human likeness. And being found in appearance as a man, He humbled Himself and became obedient to death—even death on a cross.…

Also Luke 17:33:

If you cling to your life, you will lose it, and if you let your life go, you will save it.

To conclude, the Buddha is the true light of the world. The Buddha shows the path of the best wisdom & the best human ethics. Many, such as Jesus, obviously adapted the Buddha's teachings to try to make people free. However, the Buddha is the original & the best, as the Buddha said:

Of all the paths the Eightfold Path is the best; of all the truths the Four Noble Truths are the best; of all things passionlessness is the best: of men the Seeing One (the Buddha) is the best.

Dhammapada 273