would we be able to see a total solar eclipse after about 600 million years from now?
Spectacular solar eclipses are an extreme rarity within the universe at large. They are seen on Earth because of a fortuitous combination of circumstances that are statistically very improbable. Even on Earth, spectacular eclipses of the type familiar to people today are a temporary phenomenon. Many millions of years in the past, the Moon was too close to the Earth to precisely occult the Sun as it does during eclipses today; and many millions of years in the future, it will be too far away to do so.
Due to tidal acceleration, the orbit of the Moon around the Earth becomes approximately 3.8 cm more distant each year. It is estimated that in 600 million years, the distance from the Earth to the Moon will have increased by 23,500 km, meaning that it will no longer be able to completely cover the Sun's disk. This will be true even when the Moon is at perigee, and the Earth at aphelion.
A complicating factor is that the Sun will increase in size over this timescale. This makes it even more unlikely that the Moon will be able to cause a total eclipse. We can therefore say that the last total solar eclipse on Earth will occur in slightly less than 600 million years.