Will life still be sustainable on earth when solar eclipses stop?

You have 3 completely unrelated dates in your question.

Stephen Hawking suggested that humanity could kill itself in 100 years. He's not the first to make that suggestion. He's especially concerned about global warming and artificial intelligence (by the article). Pandemic is obviously an issue today. Overpopulation. Nuclear war, asteroid strike, super-volcano, dubstep - all serious threats.

Not to get all doom and gloom but there are reasons to be concerned for humanity over the next 100 years or so. Hawking certainly has a point though I lean towards the idea of extinction being less likely, some non-extinction bumps on the road are certainly possible.

The expanding sun (and no more eclipses cause the Sun is too big), will happen, slowly. The sun increases in luminosity about 1% per 100 billion years and in diameter, maybe 1% every 200 million years. In several hundred million years our sun might be too big for Earth and Earth could start dying.

The counter-argument to this. A sun-shield wouldn't be all that hard to build. It's even being discussed today as a possible solution to climate change. Not cheap, but not impossible, especially if we have a few hundred million years to get it done.

The loss of CO2 probably will happen too though Earth's CO2 doesn't just go down neatly, it's more like an up and down with an overall downward trend. If this continues, then in maybe 100 million years (a reasonably short time), we could face as CO2 shortage. I know your article says 600 million - neither estimate is precise, but yes, a CO2 decline is probably going to happen at some point in the far future.

The good news - we know how to fix that. Burn coal. Problem solved, When we run out of coal, burn carbonate rock of other kinds. There's billions and billions of tons of carbonate rock on Earth. We can make CO2. It's not that hard. If we run out of carbonate rock (unlikely but possible) - toss some small comets rich in methane and CO2 into the ocean or onto unpopulated regions of Earth. The comet idea might not be practical but it's certainly possible and many comets are rich in CO2 and CH4.

Fun estimate. Halley's comet has a mass of about 220 trillion kg. At 280 PPM, the CO2 in the atmosphere has a mass of about 10 Halley's comets. If Halley's comet is about 25% carbon compounds, 50% water, 25% other stuff - not the worst estimate and the CH4 yields more CO2 per mass, then all you'd need is maybe 2-4 Halley's comet equivalents every 10 million years or so to maintain CO2 on Earth.

Halley's comet itself is too large but smaller comets could be tossed into unpopulated regions or into the ocean without much harm - and again, CO2 problem solved. Not impossible.

So, I think the trick is surviving Hawking's 100 year warning because the other problems are solvable with technology, at least for a time. Likely beyond 600 million years.

Stephen Hawking believed so because of technolgical reasons unrelated to geoscience, that technology and engineering will end the human existence.

Yes, if all goes well and only looking at the limiting elements "solar evolution" and "plate tectonics" life will still be there in ~600 million years. Though complexity may be drastically reduced and it will probably be limited to extremophiles in a few niches. The time of life on earth will eventually run out in the far future, but the timing is not of specifical scientific interest. Thoughts range from 500My to 1.5By, afaik.

The outline in the wikipedia article shows the principles, though I'd be careful taking such details as CO2 concentration in 600 million years too serious. Anyway, over these timescales, the sun will get hotter and plate tectonics (a transient thing anyway) will slow down and stop, interrupting regenerative and climate regulating cycles.

Statements concerning the future are, of course, highly speculative and because of that I leave it be to search for geo-scientifically relevant links on the matter. A lot of things with fundamental influence can happen much earlier (like in the coming decades).