Short answer
Not where its scriptures describe the world. The Pali texts, the oldest Buddhist scriptures, describe Sineru, the king of mountains. They make it 84,000 leagues high and 84,000 deep. A league in these texts is about seven miles, so the mountain would reach more than twice as far as the moon. In one discourse the Sun god is seized by Rahu, a titan, and set free at the Buddha's verse. In fact an eclipse of the sun is the moon's shadow. The Buddha also says the earth rests on water and the water on air, and that a great wind makes the earth quake. He adds that it quakes when a Buddha is conceived, is born or awakens. Earthquakes come from sudden slips along cracks in the earth's crust. The Prophet (ﷺ) taught that eclipses happen for no one's death or birth.
Full explanation
Why Buddhism is called scientific
Many people today see Buddhism as a science of the mind rather than a religion. The Dalai Lama, the best-known Buddhist teacher of our time, wrote that if science clearly shows a Buddhist claim to be false, Buddhists must give that claim up. So it is fair to test what the Buddhist scriptures say about the world.
The Dalai Lama writes that if scientific analysis conclusively showed certain claims in Buddhism to be false, those claims would have to be given up, and Buddhists must accept the findings of science.
The world in the Pali texts
The Pali texts describe a world system with a moon, a sun, Sineru the king of mountains, and four lands. The first, the Black Plum Tree Land, is India. A thousand such worlds make what the translator calls a "galaxy". The Buddha gives Sineru's size: 84,000 leagues long, wide, high and deep. The league of these texts, the yojana, is about seven miles.
“Ānanda, a galaxy extends a thousand times as far as the moon and sun revolve and the dazzling ones light up the quarters. In that galaxy there are a thousand moons, a thousand suns, a thousand Sinerus king of mountains, a thousand Black Plum Tree Lands, a thousand lands west of Godāvarī, a thousand lands north of Kuru, a thousand lands east of Videha, four thousand oceans, four thousand great kings, a thousand realms of the gods of the four great kings, a thousand realms of the gods of the thirty-three, of the gods of Yama, of the joyful gods, of the gods who love to create, of the gods who control what is created by others, and a thousand realms of divinity. This is called a thousandfold lesser world system, a 'galaxy'.”
“Sineru, the king of mountains, is 84,000 leagues long and 84,000 leagues wide. It sinks 84,000 leagues below the ocean and rises 84,000 leagues above it.”
Gives yojana, the league of the Pali texts, as a measure of length: as much as can be travelled with one yoke of oxen, a distance of about 7 miles.
A mountain higher than the moon
At seven miles to the league, 84,000 leagues is about 588,000 miles. The moon is on average 238,855 miles from the earth, so Sineru would rise more than twice as far as the moon. The whole earth is 7,926 miles across. A mountain sinking 84,000 leagues below the ocean would go down about 74 times the width of the earth. No such mountain exists.
“Sineru, the king of mountains, is 84,000 leagues long and 84,000 leagues wide. It sinks 84,000 leagues below the ocean and rises 84,000 leagues above it.”
Gives yojana, the league of the Pali texts, as a measure of length: as much as can be travelled with one yoke of oxen, a distance of about 7 miles.
The Moon is an average of 238,855 miles (384,400 kilometres) from the Earth. The Sun, the solar system's only star, is about 93 million miles (150 million kilometres) from the Earth, so it is about 390 times farther away than the Moon.
Earth has an equatorial diameter of 7,926 miles (12,756 kilometres).
The sun seized by Rahu
In one discourse the Sun god is seized by Rahu, a lord of the titans. The Sun god calls on the Buddha, and the Buddha tells Rahu in verse not to swallow the sun and to release it. Rahu lets the sun go and runs off in shock. This is the old Indian picture of an eclipse.
“Now at that time the Sun God had been seized by Rāhu, lord of titans. Then the Sun God, recollecting the Buddha, at that time spoke this verse:”
“Rāhu, do not swallow him as he traverses the sky. Rāhu, release my offspring, the Sun!”
“Then Rāhu, having released the Sun, rushed to see Vepacitti, lord of titans and stood to one side, shocked and awestruck.”
What an eclipse really is
An eclipse of the sun happens when the moon passes between the sun and the earth, and its shadow falls on the earth. It comes only at new moon. No being seizes the sun, and no verse makes anyone let it go.
NASA explains that a solar eclipse happens when the Moon passes between the Sun and Earth, casting a shadow on Earth that blocks the face of the Sun for observers in those places. Solar eclipses occur during the new moon phase, and lunar eclipses during the full moon phase.
Why the earth quakes
Shortly before his death, the Buddha gives his helper Ananda eight causes of great earthquakes. The first is that the earth rests on water, the water on air, and the air in space. So a great wind stirs the water, and the water shakes the earth. Others are the conception and birth of a future Buddha, and the Buddha's own awakening.
“This great earth is established on water, the water is established on air, and the air stands in space. At a time when a great wind blows, it stirs the water, and the water stirs the earth. This is the first cause and reason for a great earthquake.”
“Furthermore, when the being intent on awakening passes away from the host of joyful gods, he's conceived in his mother's womb, mindful and aware. Then the earth shakes and rocks and trembles. This is the third cause and reason for a great earthquake. Furthermore, when the being intent on awakening comes out of his mother's womb mindful and aware, the earth shakes and rocks and trembles. This is the fourth cause and reason for a great earthquake. Furthermore, when the Realized One awakens to the supreme perfect awakening, the earth shakes and rocks and trembles.”
What causes earthquakes
An earthquake is caused by a sudden slip on a fault, a crack in the earth's crust. The plates of the earth's crust move slowly and stick at their edges. When the strain overcomes the friction, the energy is released in waves, and the ground shakes. The earth does not rest on water, and a wind under it does not make it quake.
An earthquake is caused by a sudden slip on a fault. The tectonic plates are always slowly moving but get stuck at their edges by friction; when the stress overcomes the friction, the energy released travels through the earth's crust in waves and causes the shaking.
What the Qur'an says, and what it does not
The Qur'an speaks often of the sun and the moon. But it gives no distance for either, and no measure of any mountain. It says the sun may not catch up with the moon, and that each swims along in its own orbit. Qur'an 18:86 tells what the traveller Dhul-Qarnayn found when he reached the west. Qur'an 55:17 calls God Lord of the two Easts and the two Wests. Ibn Kathir, a commentator of the fourteenth century, explains these as the points where the sun rises and sets in summer and in winter.
“It is not permitted to the Sun to catch up the Moon, nor can the Night outstrip the Day: Each (just) swims along in (its own) orbit (according to Law).”
“"'See ye not how Allah has created the seven heavens one above another, "'And made the moon a light in their midst, and made the sun as a (Glorious) Lamp?”
“Until, when he reached the setting of the sun, he found it set in a spring of murky water: Near it he found a People: We said: "O Zul-qarnain! (thou hast authority,) either to punish them, or to treat them with kindness."”
“(He is) Lord of the two Easts and Lord of the two Wests:”
Explains "the two Easts and the two Wests" as the sun's rising points in summer and in winter, and its setting points in summer and in winter. He sets beside it Qur'an 70:40, "the easts and the wests", meaning the sun's rising places, which differ and move day by day, and Qur'an 73:9, "the East and the West", meaning east and west as a kind.
What the Prophet (ﷺ) said about the eclipse
The sun was eclipsed on the day the Prophet's (ﷺ) son Ibrahim died, and people said it was because of his death. He could have let them believe it. Instead he told them that the sun and the moon are not eclipsed for anyone's death or birth. And he told them to pray. The Qur'an calls the sun and the moon signs, and tells people to bow to the One who made them.
Al-Mughira ibn Shu'ba reports that the sun eclipsed on the day the Prophet's (ﷺ) son Ibrahim died, and people said it had eclipsed because of Ibrahim's death. The Prophet said that the sun and the moon do not eclipse for the death or the birth of anyone, and told them to pray and call on Allah when they see an eclipse.
“Among His Signs are the Night and the Day, and the Sun and the Moon. Do not prostrate to the sun and the moon, but prostrate to Allah, Who created them, if it is Him ye wish to serve.”
“Behold! in the creation of the heavens and the earth, and the alternation of night and day,- there are indeed Signs for men of understanding,- Men who celebrate the praises of Allah, standing, sitting, and lying down on their sides, and contemplate the (wonders of) creation in the heavens and the earth, (With the thought): "Our Lord! not for naught Hast Thou created (all) this! Glory to Thee! Give us salvation from the penalty of the Fire.”
Perspectives
Buddhism and science agree
Buddhism is a science of the mind. Its old picture of the world was the background of its time, not its teaching, and Buddhists drop any claim that science disproves.
The cosmology was the common belief of ancient India, not what the Buddha came to teach.
Rebuttal
In these texts, it is the Buddha himself who tells his monks the size of Sineru, and tells Ananda the causes of earthquakes. Elsewhere he says he speaks of what he has known and seen for himself.
The thousandfold "galaxy" of the Pali texts shows that the Buddha knew of many galaxies.
Rebuttal
The text calls it a "thousandfold lesser world system". The word "galaxy" is the translator's. Each of its worlds has its own sun, its own moon, its own Sineru, and its own Black Plum Tree Land, the land of India. It is the ancient Indian world copied a thousand times, not a galaxy of stars.
Buddhists give up any claim that science disproves. So the old picture of the world does not count against Buddhism.
Rebuttal
Then Buddhists must give up words that the Pali texts give as the Buddha's own, about the earth, the sun and earthquakes. A teacher who was wrong about things anyone can check needs some other proof. That is so for rebirth and nirvana, which no one can check.
The Islamic perspective
Where the Pali texts speak of the world, they give measures and causes that the earth and sky show to be wrong. The Qur'an gives none. And the Prophet (ﷺ) corrected the belief that an eclipse marks a death.
The Pali texts make Mount Sineru 84,000 leagues high and 84,000 deep. At about seven miles to the league, that is more than twice the distance to the moon.
In the Pali texts Rahu seizes the sun and lets it go at the Buddha's verse. An eclipse of the sun is the moon's shadow.
The Buddha says the earth rests on water and quakes when a great wind stirs it, and when a Buddha is born or awakens. Earthquakes come from sudden slips on faults.
The Prophet (ﷺ) taught that the sun and the moon are not eclipsed for anyone's death or birth. And the Qur'an calls them signs of their Creator.