I do not like to call anything discovered now as 'scientific fact' as any fact which is established now can be disproved in the future. (For eg. quantities like mass, time etc.. were considered absolute in the past and now they are considered relative). So, I like to use the term 'scientific concepts' for the present discoveries. There can be many scientific facts which are in Hindu scriptures and here are few concepts described in relation with modern scientific discoveries. (What I mean is following concepts can be derived after reading those quoted verses):
1) Concept of Gravity:
Vaishesika Sutras discuss about Gravity which I discuss in my answer here. Relevant quotes from that answer are:
आत्मकर्म हस्तसंयोगाश्च । संयोगभावे गुरुत्वात्पतनम।
नोदनाद्यभिषोः कर्म तत्कर्मकारिताच्च संस्कारादुत्तरं तथोत्तरमुत्तरं च । संस्काराभावे गुरुत्वात्पतनम।
अपां संयोगाभावे गुरुत्वात्पतनम। द्रवथ्वास्यन्दनम् ।
Action of body and it's members is also from conjunction with the hand. In the absence of conjunction falling results from Gravity. The first action of arrow is from impulse; the next is resultant energy produced by the first action, and similarly the next next. In the absence of resultant/propulsive energy generated by action, falling results from Gravity. The falling of water in absence of conjunction is due to Gravity. Flowing results from fludity.
2) Concept of Time Dilation:
We can find various stories about Time Dilation in Hindu scripture. The most famous is the story of Kakudami which I discuss here. They sit in BrahmaLoka for only about 20 minutes and millions if years passes in the Earth. Relevant verses from Srimad Bhagvatam 9.3 is as:
आवर्तमाने गान्धर्व स्थितो ’लब्ध-क्षणः क्षणम् ।
तद्-अन्त आद्यम् आनम्य स्वाभिप्रायं न्यवेदयत्।।
तच् छ्रुत्वा भगवान् ब्रह्मा प्रहस्य तम् उवाच ह।
अहो राजन् निरुद्धास् ते कालेन हृदि ये कृताः।।
तत् पुत्र-पौत्र-नप्तॄणां गोत्राणि च न शृण्महे।
कालो ’भियातस् त्रि-णव-चतुर्-युग-विकल्पितः।।
When Kakudmi arrived there, Lord Brahma was engaged in hearing musical performances by the Gandharvas and had not a moment to talk with him. Therefore Kakudmi waited, and at the end of the musical performances he offered his obeisances to Lord Brahma and thus submitted his long-standing desire. After hearing his words, Lord Brahma, who is most powerful, laughed loudly and said to Kakudmi: O King, all those whom you may have decided within the core of your heart to accept as your son-in-law have passed away in the course of time. Twenty-seven catur-yugas have already passed. Those upon whom you may have decided are now gone, and so are their sons, grandsons and other descendants. You cannot even hear about their names.
3) Concept of Multiverses:
The concept of Multiverse has not been practically proved by modern science. But still it is an interesting concept in Modern science also. Hindu scriptures talk about multiverses. For instance as I discuss in my answer here, Srimad Bhagvatam in 10.14.11 talks of multiverses:
वाताध्व-रोम-विवरस्य च ते महित्वम् ।।
What am I, a small creature measuring seven spans of my own hand? I am enclosed in a potlike universe composed of material nature, the total material energy, false ego, ether, air, water and earth. And what is Your glory? Unlimited universes pass through the pores of Your body just as particles of dust pass through the openings of a screened window.
Similarly as discussed in answer here Devi Bhagvatam also states presence of other Brahma, Vishnu and Rudras in other universes:
Then the car, where we were situated, began to get up high in the sky, and in the twinkling of an eye, we reached Brahmâ loka, that is saluted by all the Devas. There S'ambhu and Kes'ava were greatly bewildered to see Brahmâ of that place. In the council hall of Brahmâ, the Vedas with their Angas, the serpents, hills, oceans and rivers were seen. Seeing all these, Visnu and Mahes'vara asked me :-- “O Four-faced one! Who is this eternal Brahmâ? I replied :-- I do not know who is this Brahmâ? Who am I? and who is He? why has this error come over me? You, too, also are gods so you can better ponder over it.”
Similarly as quoted in the same answer, chapter 10 'Shiva swallowing poison' of Skandha Purana also states:
He (Ganesha) saw crores of spherical Cosmic Eggs like so many atoms. They were getting merged and dissolved in Mahesha who was in linga form.
Similarly as I discuss in my answer here Ramcharitmanas also mentions Kakabhusundi visiting multiple universes and even visiting himself:
In all these universes I saw myself in every universe. Each universe had its own Ayodhya with its own Sarayµu and its own men and women. And listen, dear Garuda : Ramas parents Dasaratha and Kausalya as well as Ramas brothers, Bharata and others, were all different in each universe. In each such universe I witnessed the descent of Rama as well as the infinite variety of His childish sports.
4) Concept of sphericity of Earth and rising-setting of Sun:
As I discuss in my answer here, Srimad Bhagvatam 5.21.9 states:
तत्रत्यानां दिवस-मध्यङ्गत एव सदादित्यस् तपति सव्येनाचलं दक्षिणेन करोति; यत्रोदेति तस्य ह समान-सूत्र-निपाते निम्लोचति यत्र क्वचन स्यन्देनाभितपति तस्य हैष समान-सूत्र-निपाते प्रस्वापयति तत्र गतं न पश्यन्ति ये तं समनुपश्येरन्.
The living entities residing on Sumeru Mountain are always very warm, as at midday, because for them the sun is always overhead. Although the sun moves counterclockwise, facing the constellations, with Sumeru Mountain on its left, it also moves clockwise and appears to have the mountain on its right because it is influenced by the dakṣiṇāvarta wind. People living in countries at points diametrically opposite to where the sun is first seen rising will see the sun setting, and if a straight line were drawn from a point where the sun is at midday, the people in countries at the opposite end of the line would be experiencing midnight. Similarly, if people residing where the sun is setting were to go to countries diametrically opposite, they would not see the sun in the same condition.
Regarding the rising and setting of Sun, Aitreya Brahmana of RigVeda 3.44 states:
The Sun does never set nor rise. When people think the Sun is setting (it is not so). For after having arrived at the end of the day it makes itself produce two opposite effects, making night to what is below and day to what is on the other side…Having reached the end of the night, it makes itself produce two opposite effects, making day to what is below and night to what is on the other side. In fact, the Sun never sets….”
Similarly as discussed in the same answer Yoga Vasistha also states:
10 Vasishta said:—
There is only one space enveloping all things. The worlds seen in the infinite and indiscernible womb of emptiness are like worms moving on the surface of water.
11 All these bodies that move about in the world by their lack of freedom are thought to be up and down relative to our position on earth.
12 So when there are ants on an earthen ball, all its sides are reckoned below that are under their feet, and those as above which are over their backs.
13 Such is this ball of earth in one of these worlds, covered by vegetables and animals moving on it, and by gods, demons and men walking upon it.
5) Concept of Universe as being Projection in space-time:
As I discuss in my answer here Vedas state Universe as being appearance in space time.
पूर्ण कुम्भोदिः काल आहितस्तं वै पश्यामो बहुदा नु सन्तः । [AtharvaVeda 19.53.3]
A full pot has been placed in Time and it is that which we see manifoldly.
Also, in the Dakshinamurty stotram composed by Adi Shankara he states:
बीजस्याऽन्तरिवाङ्कुरो जगदिदं प्राङ्गनिर्विकल्पं पुनः
मायाकल्पितदेशकालकलना वैचित्र्यचित्रीकृतम् ।
This World is Like a Sprout of a Seed Within which transforms what is Changeless state Before, appear Again as Space and Time, and endless Varieties of Pictures over it; all due to the Creation of Maya.
6) Concept of Cyclic Nature of Time:
As I discuss in my answer here Rig Veda states about Cyclical Nature of Time and Creation:
सूरय्याचन्द्रमसौ धाता यथा पूर्वमकल्पयत् I
दिवं च पृथ्वीं च अंतरिक्षमथो स्वः II RigVeda 10.190.3 II
"The Ordainer created the sun and moon like those of previous cycles. He formed in order Heaven and Earth, the regions of the air, and light."
7) Concept of Speed of Light:
As I discuss in my answer here Sayanacharya while commenting on RigVeda on a verse related to Sun, quotes from Taittariya Brahmana of Yajurveda.
तथा च स्मर्यत योजनानां सहस्रम् द्वे द्वे शते द्वे च योजने । एकेन निमिषार्धेन क्रममाण नमोऽस्तु ते ॥
It is remembered, [O Sun] bow to you, you who traverse 2,202 yojanas in half a nimesha.
And on converting the above quote with respect to modern figures as done in that answer we get approximate value to speed of light.
8) Concepts of One being essence of another:
There are some scientific concepts in present day which are related with the nature of Observer. Ie. For the existence of something an observer is required. for eg. If we say fire exists there then there should be an observer to acknowledge the existence of fire (It is related to quantum physics concept and simple common sense can't solve it). Another simple example is Schrodingers Cat experiment related to observer (and also Many World Theory which comes to solve it). In conclusion the concept is if Fire is the essence of living beings then living beings are also essence of fire. Ie. They have mutual existence.
A similar Knowledge is taught in Brihadaranyaka Upanishad which is also called Madhu Vidya. A literal translation can be 'Knowledge of Honey' but it actually means 'Knowledge of Essence'. (Just like Honey is the essence prepared by Bees from flowers). It starts from the Brihadaranyaka Upanishad 2.5.1. Some of it's verses are:
II-v-1: This earth is (like) honey to all beings, and all beings are (like) honey to this earth. (The same with) the shining immortal being who is in this earth, and the shining, immortal, corporeal being in the body. (These four) are but this Self. This (Self-knowledge) is (the means of) immortality; this (underlying unity) is Brahman; this (knowledge of Brahman) is (the means of becoming) all.
II-v-2: This water is (like) honey to all beings, and all beings are (like) honey to this water. (The same with) the shining immortal being who is in this water, and the shining, immortal being identified with the seed
in the body. (These four) are but this Self. This (Self-knowledge) is (the means of) immortality; this (underlying unity) is Brahman; this (knowledge of Brahman) is (the means of becoming) all.
II-v-3: This fire is (like) honey to all beings, and all beings are (like) honey to this fire. (The same with) the shining immortal being who is in this fire, and the shining, im
mortal being identified with the organ of speech in the body. (These four) are but this Self. This (Self-knowledge) is (the means of) immortality; this (underlying unity) is Brahman; this (knowledge of Brahman) is (the means of becoming) all.
II-v-4: This air is (like) honey to all beings, and all beings are (like) honey to this air. (The same with) the shining immortal being who is in this air, and the shining, immortal being who is the vital force in the body. (These four) are but this Self. This (Self-knowledge) is (the means
of) immortality; this (underlying unity) is Brahman; this (knowledge of Brahman) is (the means of becoming) all.
II-v-5: This sun is (like) honey to all beings, and all beings are (like) honey to this sun. (The same with) the shining immortal being who is in this sun, and the shining, immortal being identified with the eye in the body. (These four) are but this Self. This (Self-knowledge) is (the means of) immortality; this (underlying unity) is Brahman; this
(knowledge of Brahman) is (the means of becoming) all.
Thus the summary of the Madhu Vidya is all the things in this Universe are interelated and everything is the essence of Everything.For eg. If air is the essence of all beings, then all beings are also the essence of air. Madhu Vidya also states:
II-v-13: This human species is (like) honey to all beings, and all beings are (like) honey to this human species. (The same with) the shining immortal being who is in this human species, and the shining, immortal being identified with the human species in the body. (These four) are but this Self. This (Self-knowledge) is (the means of) immortality; this (underlying unity) is Brahman; this (knowledge of Brahman) is (the means of becoming) all.
And the Madhu Vidya Concludes with an important verse:
II-v-15: This Self, already mentioned, is the ruler of all beings, and the king of all beings. Just as all the spokes are fixed in the nave and the felloe of a chariot-wheel, so are all beings, all gods, all worlds, all organs and all these (individual) selves fixed in this Self.
Thus the concept is just as this Universe is an essence for me similarly I'm the essence of this whole Universe. (Note That Madhu Vidya is one of the most important Vidyas and it was also used to revive the Deads also by the Gods and Rishis)
The above I have quoted are some verses/stories which can be related to modern scientific discoveries. There are still much more things which I have to write here I will try to continue updating the answer regularly.