Veda is Apaurusheya. All other Pramanas (means of knowledge) with different Aastika schools of thought accept Veda as the authoritative text and with different means arrive at the meaning of Vedas. But as for the validity of Veda (as far as I know) they do not talk. Because if we raise the question on the authority of Veda itself, then we cannot take anything as Pramana. But since you asked, I will quote only one sloka I know which I hope should serve the purpose.
In Bhagavad Gita, the Lord says
“Shruti:smriti: mamaiva AgyA-yastAm ullanghya vartatE AgyAcchEdI mama
drOhI mama bhaktOpi na Vaishnava:”
Sruti and Smruti are my commands. He who violates them is a traitor.
Edit: I searched for the source of the above sloka which is not Bhagavadgita. So I am editing the answer to explain that it is difficult to prove the authority of the Vedas.
Vedas are Apaurusheya meaning they are not created by or verbal testimonies of human being or any other being, but they are revealed to Rishis and Gods who were in penance. The Lord himself says in Gita as तस्माच्छास्त्रं प्रमाणं ते कार्याकार्यव्यवस्थितौ (BG-Ch16-Verse24) meaning śāstra (vedas) is the means of knowledge for you (Arjuna), in the determination of what is to be and not to be done. You can take this as one very valid argument because the lord himself declared that they are the authority for everything.
Jagadguru Shankaracharya of Sringeri HH Sri Bharati Teertha Mahaswami explains in his Anugraha Bhashanam explains as
When we mention the term “Shastras”, we only refer to the Vedas. The
word “Shastram” is derived from the verse – “शास्ति इति शास्त्रम्”.
Only the Vedas can assertively say what needs to be done. It is very
difficult to question what is mentioned in the Vedas. The Dharma
Shastras mention that Vedas are the supreme authority on Dharma in the
verse – वेदोऽखिलो धर्ममूलम्. Dharma refers to something that brings us
good and Vedas list down the things that bring us good and the things
that we are not expected to do. We should adhere to what is mentioned
in the Shastras. The philosophers of yore have all stated – यच्छब्द
आह, तदस्माकं प्रमाणम् – “Shabda” here refers to the Vedas. What the
Vedas state is the authority for us.
There is a story in Tripura Rahasyam regarding the faith in the Shastras which goes like this
Once there lived a saint, by name Kausika, on the Sahya Hill near the
banks of the Godavari. He was serene, pure, pious, having knowledge of
the Supreme Truth. Several disciples attended on him. Once when the
master had gone out, the disciples started to discuss philosophy,
according to their own lights. There appeared on the scene a Brahmin
of great intellect and wide learning, Soonga by name, who successfully
refuted all their arguments by his skill in logic. He was a man
without faith and without conviction, but an able debater. When they
said that the truth must be ascertained by reference to some standard,
he argued on the basis of an unending series of standards and refuted
them. He rounded off his speech with the following: Listen, you
Brahmins, standards are not applicable for ascertaining merits or
demerits and so arriving at the truth. For erroneous standards are no
good as tests. To start with, their correctness must be established.
Other standards are required to check them. Are they in their turn
infallible? Proceeding in this way, no finality can be reached.
Therefore no tests are possible. Ascertainment of Truth being
impossible without being tested, nothing can therefore be Truth. This
enunciation itself cannot be true, nor the enunciator either. What
then is the decision arrived at? That all are nothing, void. This too
cannot be supported by reliable facts; hence, the statement that all
are void ends in void also. Hearing his discourse, some of them were
impressed by the force of Soonga’s logic and became scholiasts of the
void. They got lost in the maze of their philosophy. The
discriminating ones among the hearers placed Soonga’s arguments before
their master and were enlightened by him. Thus they gained peace and
happiness. Therefore, beware of arid polemics parading as logic. Use
it in the manner in which the holy books have done. That way lies
The person narrating this story is a jivanmukta in this whose story is narrated by Lord Dattatreya himself to Lord Parasurama.
The Pramanas which were accepted by schools like Nyaya, vaiseshika, Mimamsa, Sankhya etc., were called āstika schools because they accept veda as authority and have different views regarding the ultimate truth. Also the whole Smritis, Puranas, Dharma Shastras came inorder to explain the heart of veda.
Kanchi Paramachaya HH Jagadguru Sri Chandrasekhara Saraswati explains as (Hindu Dharma - The Universal Way of Life)
The Puranas are the magnifying glass of the Vedas. The principles and
rules of dharma that are briefly dealt with in the Vedas are enlarged
or elaborated upon in them in the form of stories. (Magnifying Glass
of the Vedas)
The Vedas are the root of this dharma, its fountainhead. But the rites
and duties are not given in an orderly form in the Vedas, nor is the
procedure for works laid down in detail. Of the Vedas that are
infinite we have obtained only a very small part. And we do not
comprehend fully the meaning of many of the passages even of this
small part. The Dharmasutras (by Apastamba, Gautama and others) are
terse statements and are so according to the very definition of the
term "sutras". The dharmasastras (by Manu, Yagnavalkya, Parasara and
others) are called Smrtis and are in verse and detail in treatment.
Their basis, however, is constituted by the Vedas. The function of
Dharmasastra is to analyse and explicate the sutras of Kalpa which
have to some extent systematised the Vedic rules and injunctions. If
Kalpa gives instructions about the constructions of the Vedic altar,
of houses, etc, Dharmasastra provides a code of conduct embracing all
human activities. "Smrti" is what is remembered. "Vismrti" is
insanity. Manu observes :"There is Smrti for the Vedas in the form of
notes. The sages who had a profound understanding of the Vedas have
brought together the duties and rites (dharma and karma) mentioned in
them in the form of notes and they constitute the Smrtis. They are
written in a language that we can easily understand. Read them. They
tell you about your in detail, the do's and don'ts, and how the rites
are to be performed. " (Realising the Ideals of the Puranas)
The best testimony to the claim that the Smrtis are founded on the
Vedas is provided by the words of mahakavi(great poet). The greatest
of the mahakavis, Kalidasa, makes a reference to the Smrtis in his
Apamsulanam dhuri kirtaniya
-Raghuvamsam, 2. 2
So Sudhaksina followed in the "hoof steps" of Nandini like the Smrtis
following the meaning of the Vedas. Also, like the Smrtis not going in
the entire way with the Vedas, she did not go all the distance with
the cow. The idea is that the Smrtis do not repeat all that is said in
the Vedas. They are "notes from memory", but they truthfully follow
the Vedas in their meaning. They do not, of course, represent all
thousands of mantras of the scriptures but, all the same, they tell us
how to make use of the Vedas.
"Sudhaksina with her pure antah-karana followed her husband and,
without deviating even a little, walked along the path of the dust
raised by Nandini's hoofs". Having said so much, Kalidasa thought he
must bring in a good simile for Sudhaksina following the cow dust and
it occurred to him in a flash: "Like the Smrtis following faithfully
the meaning of the Vedas. " (The Source of Smritis is the Vedas)
Also in Ribhu Gita, Lord Siva says
श्रुतिरियं कर्तृत्ववादं वदत् meaning śruti (vedas) alone are the
authority and guidance for all.
So, It is very difficult to prove the authority of the vedas as stated by Sringeri Jagadguru. Also if the schools whose sole existence was possible just because they were āstika gave arguments for vedas' authority, then how valid are their arguments ?
So always Vedas are the pramana whose authority cannot be questioned (though it might sound harsh) and is difficult to be proved based on means like Pratyaksha, Anumana.
Hope this helps you.