Short answer
No one can show that it is. Hindus hold that Krishna spoke the Gita to Arjuna before a great battle, just before the present age began. The astronomers' treatises date that age from 3102 BC. But the Gita has come down inside the Mahabharata, and no unbroken chain of named teachers links its words to Krishna. The epic's own opening speaks of a shorter Bharata and a much longer one. V. S. Sukthankar, the Indian scholar who led its critical edition, called the text fluid from the start. He said its original shape cannot be rebuilt. The Gita itself says its teaching was once lost over time. The Qur'an came through a man known to history, and copies survive from close to his lifetime.
Full explanation
The Hindu teaching: the Lord's own words
The Bhagavad Gita is part of the Mahabharata, the great epic. It belongs to smriti, the remembered tradition. In it Krishna teaches the warrior Arjuna on the field of Kurukshetra, just before the battle. Sanjaya, the blind king's charioteer, relates the dialogue to the king. Most Hindus hold the Gita to be the teaching of the Lord himself, and many read it daily. The Vishnu Purana says when this happened: "The day that Krishńa shall have departed from the earth will be the first of the Kali age". The Kali age is the present age of the world. Classical Hindu treatises on astronomy, such as the Surya Siddhanta, reckon its start from midnight between 17 and 18 February 3102 BC.
“The day that Krishńa shall have departed from the earth will be the first of the Kali age, the duration of which you shall hear; it will continue for 360,000 years of mortals.”
In his notes to the first chapter, Burgess explains that the Surya Siddhanta, a classical Hindu treatise on astronomy, and the treatises that follow its method reckon everything from the start of the present age, the Kali yuga. They set that start at midnight on the meridian of Ujjain, between 17 and 18 February 3102 BC.
How the epic says it was passed on
The Mahabharata opens by telling how it was handed down. Vyasa, to whom the epic is credited, taught it to his pupil Vaisampayana. Vaisampayana recited it at the sacrifice of King Janamejaya. The narrator, Sauti, heard it there and recites it again. The same opening names more than one size of the work. It says that Vyasa's Bharata, without its episodes, was "originally in twenty-four thousand verses". Then it speaks of "another compilation, consisting of six hundred thousand verses", of which "one hundred thousand" are known among people. So the epic itself speaks of a shorter Bharata and a far longer one.
“Vyasa executed the compilation of the Bharata, exclusive of the episodes, originally in twenty-four thousand verses; and so much only is called by the learned as the Bharata.”
“After that he executed another compilation, consisting of six hundred thousand verses. Of these, thirty hundred thousand are known in the world of the Devas; fifteen hundred thousand in the world of the Pitris; fourteen hundred thousand among the Gandharvas, and one hundred thousand in the regions of mankind.”
V. S. Sukthankar, the Indian scholar who led the critical edition of the Mahabharata at the Bhandarkar Oriental Research Institute in Poona, writes that the epic's text was fluid from the start. It was first recited freely from memory, as faithfully as each reciter could manage. Sauti, who recited it again, had heard it only once, at that first recitation. The strict safeguards used for the Vedic texts never existed, as far as one knows, for the epics. The tradition itself, he notes, says that Vyasa's five pupils published five separate versions. He concludes that it is useless to think of rebuilding the text in its literally original shape. The edition can only aim at the oldest form of the text that the manuscripts allow.
What the editors of the epic found
In the last century, scholars at the Bhandarkar Oriental Research Institute in Poona compared the Mahabharata's manuscripts from across India. Their leader, V. S. Sukthankar, wrote that the text was fluid from the start. It was first recited freely from memory, as faithfully as each reciter could. The strict safeguards used for the Vedas never existed, as far as one knows, for the epics. He concluded that it is useless to try to rebuild the text in its original shape. His edition could only reach the oldest form the manuscripts allow. Earlier, the American scholar E. W. Hopkins dated the epic's stages. He placed the "remaking of the epic with Krishna as all-god" between 200 BC and AD 200. Hopkins called his dates "provisional". The point does not hang on any one date. Even on the traditional dating, no chain of named teachers covers the time between Krishna and the written text.
V. S. Sukthankar, the Indian scholar who led the critical edition of the Mahabharata at the Bhandarkar Oriental Research Institute in Poona, writes that the epic's text was fluid from the start. It was first recited freely from memory, as faithfully as each reciter could manage. Sauti, who recited it again, had heard it only once, at that first recitation. The strict safeguards used for the Vedic texts never existed, as far as one knows, for the epics. The tradition itself, he notes, says that Vyasa's five pupils published five separate versions. He concludes that it is useless to think of rebuilding the text in its literally original shape. The edition can only aim at the oldest form of the text that the manuscripts allow.
“Remaking of the epic with Krishna as all-god, intrusion of masses of didactic matter, addition of Puranic material old and new; multiplication of exploits, 200 B. C. to 100-200 A. D.”
“In the case of these more precise dates there is only reasonable probability. They are and must be provisional till we know more than we know now.”
The Gita says its own teaching was lost
The Gita gives its own history. Krishna tells Arjuna that he first taught this way of devotion to the sun. The sun taught it to Manu, and Manu to the king Ikshvaku. "Coming thus by steps, it became known to royal sages. But, O terror of (your) foes! that devotion was lost to the world by long (lapse of) time." So Krishna teaches it again. By the Gita's own account, then, a teaching held to be from God (﷾) can be lost while people pass it down. Islam does not hold the Gita to be revealed. But it agrees that earlier revelations were lost or changed as people passed them on. That is why it holds that God (﷾) sent a last book, and promised to guard it Himself.
“This everlasting (system of) devotion I declared to the sun, the sun declared it to Manu, and Manu communicated it to Ikshvâku. Coming thus by steps, it became known to royal sages. But, O terror of (your) foes! that devotion was lost to the world by long (lapse of) time.”
“We have, without doubt, sent down the Message; and We will assuredly guard it (from corruption).”
How the Qur'an reached us
The Qur'an came through a man whose life is known from history. Muhammad (ﷺ) lived from about 570 to 632. He recited the Qur'an aloud, in public, over twenty-three years, and his companions learned it by heart. After his death, Abu Bakr had it gathered from the written pieces and from those who knew it by heart. Uthman later had copies made and sent to the provinces. Two early leaves at the University of Birmingham are carbon dated to between 568 and 645. Apart from small points of spelling, their text is the text Muslims recite today. Many Muslims still learn the whole Qur'an by heart, from teachers who learned it the same way. God (﷾) says: "We have, without doubt, sent down the Message; and We will assuredly guard it (from corruption)" (15:9).
Zayd ibn Thabit reports that after many who knew the Qur'an by heart were killed at Yamama, Abu Bakr asked him to gather the Qur'an into one volume, reminding him that he used to write down the revelation for the Messenger of Allah. Zayd collected it from written pieces and from the memories of men, and the finished copy passed from Abu Bakr to Umar and then to Hafsa.
Anas reports that Hudhayfa ibn al-Yaman, alarmed at how the soldiers of Syria and Iraq differed in reciting the Qur'an, urged Uthman to save the community before it differed over its Book as the Jews and Christians had. Uthman borrowed the collected sheets kept by Hafsa, had Zayd ibn Thabit and three men of Quraysh copy them, telling the three that wherever they differed with Zayd they should write in the dialect of Quraysh, in which the Qur'an was revealed, returned the sheets to her, sent a copy to every province, and ordered all other written Qur'anic material to be burned.
Two parchment leaves carrying parts of Surahs 18 to 20 in early Hijazi script. In 2015 the Oxford Radiocarbon Accelerator Unit dated the parchment to between 568 and 645 CE, with 95.4 per cent probability, a span that covers the Prophet's (ﷺ) lifetime (c. 570-632). Apart from small points of spelling, the text on the leaves is the text of those surahs as the Qur'an has them today.
A Harvard historian of religion studies how scriptures live in speech and memory. In the part titled "An Arabic Reciting: Qur'an as Spoken Book", he describes the Qur'an as above all a recited scripture, whose name means recitation, learned by heart and heard aloud more than read silently.
“We have, without doubt, sent down the Message; and We will assuredly guard it (from corruption).”
The same question for every scripture
Muslims believe in Noah, Abraham and Moses because the Qur'an names them, not because of digging. The Qur'an also says God (﷾) sent messengers to every people (16:36). So the fair question is not whether a figure can be proven by digging. It is whether the book we rely on reached us as it was given. The Qur'an asks to be tested this way. It says it would hold much contradiction if it were not from God (4:82). It challenges anyone in doubt to produce a chapter like it (2:23). Muslims invite every Hindu reader to read it with care, and to consider the man it calls the last of the prophets (33:40).
“For We assuredly sent amongst every People a messenger, (with the Command), "Serve Allah, and eschew Evil": of the People were some whom Allah guided, and some on whom error became inevitably (established). So travel through the earth, and see what was the end of those who denied (the Truth).”
“Do they not consider the Qur'an (with care)? Had it been from other Than Allah, they would surely have found therein Much discrepancy.”
“And if ye are in doubt as to what We have revealed from time to time to Our servant, then produce a Sura like thereunto; and call your witnesses or helpers (If there are any) besides Allah, if your (doubts) are true.”
“Muhammad is not the father of any of your men, but (he is) the Messenger of Allah, and the Seal of the Prophets: and Allah has full knowledge of all things.”
The replies, answered
Bhagavad Gita 4:1-2 — chapter 4, verses 1-2
“This everlasting (system of) devotion I declared to the sun, the sun declared it to Manu, and Manu communicated it to Ikshvâku. Coming thus by steps, it became known to royal sages. But, O terror of (your) foes! that devotion was lost to the world by long (lapse of) time.”
Cited for
The Gita says its teaching came down "by steps" from the Lord through the royal sages. That is an unbroken chain.
The response
The same verse goes on: "that devotion was lost to the world by long (lapse of) time." Krishna then teaches it again to Arjuna. After that, no chain of named teachers runs from Arjuna or Sanjaya to the scribes who wrote the epic down.
Where it lands — The Gita's own chain broke once, by its own account, and no named chain runs from Kurukshetra to the manuscripts.
Bhaktivedanta Swami Prabhupada, Bhagavad-gita As It Is, the disciplic succession
The introduction of the Gita edition of the International Society for Krishna Consciousness (ISKCON) says that the book is received through a disciplic succession. Its numbered list begins: Krishna, Brahma, Narada, Vyasa, and then Madhva, Padmanabha, Nrhari, Madhava, Akshobhya and Jaya Tirtha, and it runs on to Prabhupada's own teacher.
Cited for
ISKCON's Gita names its chain of teachers: Krishna, Brahma, Narada, Vyasa, Madhva and onwards. That is the chain this page says is missing.
The response
The list moves from Vyasa to Madhva in one step. Madhva lived in the thirteenth century AD. A list of honoured teachers of a school is not a record of who heard the text from whom across the many centuries between.
Where it lands — The named succession leaves a gap of many centuries before Madhva.
E. W. Hopkins, The Great Epic of India (1901), p. 398 — how sure the dates are
“In the case of these more precise dates there is only reasonable probability. They are and must be provisional till we know more than we know now.”
Cited for
Hopkins was an American writing in 1901. His dates are guesses.
The response
Hopkins said so himself: his dates were "provisional". But the case does not rest on him. The Indian editor of the critical edition found the text fluid from the start. And the epic's own opening names a shorter and a longer work.
Where it lands — Take away every Western date and the problem remains: there is no known chain from Krishna to the text.
Qur'an 16:36 — Surat al-Nahl (The Bee), verse 36
“For We assuredly sent amongst every People a messenger, (with the Command), "Serve Allah, and eschew Evil": of the People were some whom Allah guided, and some on whom error became inevitably (established). So travel through the earth, and see what was the end of those who denied (the Truth).”
Cited for
Muslims cannot prove that Noah or Moses existed either. Why ask Hindus for proof of Krishna?
The response
This page does not ask for stones. Muslims believe in Noah and Moses because the Qur'an names them. The question is the same for both books: did the words reach us as they were given? The Qur'an came through a known man and was written and learned by heart from the start.
Where it lands — The fair test is how a scripture reached us, and the Qur'an asks to be tested.
Perspectives
The Lord's word, faithfully kept
Krishna, the Lord, spoke the Gita to Arjuna, and the tradition has kept his words. Teachers passed it from guru to pupil, as the Vedas were passed down. Western dates are guesses, and the tradition is older than any manuscript.
The view of most Hindus, who hold the Gita to be the teaching of the Lord Krishna himself
The Gita is the Lord's own word, so it cannot be lost.
Rebuttal
The Gita itself says the Lord's teaching "was lost to the world by long (lapse of) time" (4:2). So by its own account, a teaching from God (﷾) can be lost while people pass it on.
The Gita came down by the unbroken chain of teacher and pupil (guru-shishya parampara), as the Vedas did. The Gita itself says its teaching came "by steps".
Rebuttal
Sukthankar led the critical edition. He found that the strict safeguards of Vedic recitation never existed for the epics. The epic was recited freely. And the tradition says Vyasa's five pupils published five versions.
The Gita comes down through a named disciplic succession (parampara), a line of teachers. It runs from Krishna to Brahma, Narada, Vyasa, Madhva, and on to teachers today.
Rebuttal
The list goes straight from Vyasa to Madhva, who lived in the thirteenth century AD. It honours the teachers of a school. It does not show who passed the text to whom over the many centuries in between.
Western scholars such as Hopkins date the epic late because they doubt Hindu tradition.
Rebuttal
The Indian scholar who led the critical edition came to the same view of the text. It was fluid from the start, and its original shape is beyond reach. And the epic's own opening speaks of a Bharata of 24,000 verses, and of a far longer work.
Sukthankar spoke of the whole epic. The Gita, its holiest part, was kept with special care.
Rebuttal
The Gita has come down only inside the epic, and Sukthankar's edition is of the epic, Gita included. Careful copying in later centuries cannot show what was said at Kurukshetra. And the Gita itself says its teaching was once lost (4:2).
Tradition dates Krishna to the start of the Kali age, 3102 BC. So the Gita is older than any manuscript.
Rebuttal
Then the gap between Krishna and the written text is longer, not shorter. No chain of named teachers fills those thousands of years.
The Islamic perspective
The question to ask of any scripture is how its words reached us. The Gita has no chain of named teachers back to Krishna, and its epic was fluid from the start. The Qur'an came through a known man. It was learned by heart and written down from the beginning. And God (﷾) promised to guard it.
The Gita says its own teaching was lost over time and had to be taught again.
The editor of the epic's critical edition found its text fluid from the start. He found its original shape beyond reach.
The Qur'an came through Muhammad (ﷺ), a man of recorded history. It was learned by heart, gathered under Abu Bakr and copied under Uthman, and early copies match it.
God (﷾) promised to guard the Qur'an, and the Qur'an asks every reader to test it.