Will the stock market continue to grow forever?

"The stock market" may not grow "forever". There will be growth in the stock market, though.

The stock market is a positive-sum game, since it is driven in large part by the profits earned by the companies. This doesn't mean that any individual stock will go up forever, it doesn't mean that any given index will go up forever, and it doesn't mean there won't be periods when the market as a whole drops. But it is reasonable to expect that long-term investing in the market as a whole will continue to return profits that reflect the success of companies invested in. Historically, that return has averaged about 8%; future results may be different and exact results will depend on exactly when and how you invest.

Re "what about Japan, which has been flat over 30 years": Market being flat doesn't mean individual companies may not be growing strongly. Picking stocks may become more important, and we might need to relearn to focus on dividends rather than being so monomaniacal about growth (dividends are not reflected in the indices, please note), but there will be money to be made. How much, and how much effort is required to get it, and whether the market offers the best available bets, deponent sayeth not.

Past results are no guarantee of future returns, and your results may be better or worse than average. You should be diversified into bonds and such anyway, rather than only in the stock market.

Yes! Look at any graph or chart covering the last 100 years. The graph goes up. It will continue to grow unless there is an extinction event and the population gets reduced. Corporations will continue to grow to meet the needs of the ever expanding population.

The answer to your question depends on what you mean when you say "growth". If you mean a literal increase in the aggregate market capitalization of companies, across the entire market, then, no, this sort of growth is not possible without concomitant economic growth. The reason why is that the market capitalization of each company is proportional to its gross revenue, and the sum of all revenue from selling "final goods" (i.e., things purchased and used by consumers) is, apart from a few technicalities, the definition of GDP. The exact multiplier might fluctuate up or down depending on investors' expectations about how sales will grow or decline going forward, but in a zero-growth economy this multiplier should be stable over the long run. It might, however, still fluctuate over the short term, but more about that in a minute.

Note that all of this applies to aggregate growth across all firms. Individual firms can still grow, of course, but as they must do this by gaining market share from other companies such growth would be balanced by a decline for some other firm. Also, I've assumed zero net exports (that's one of the "technicalities" I mentioned above) because obviously you could have export-driven growth even if the domestic economy were stationary.

However, often when people talk about "growth" in the market, what they really mean is "return". That is, how much does your investment earn for you. This isn't really the same thing as growth, but people often think of it that way, particularly in the saving phase of their investing career, when they are reinvesting their returns, and therefore their account balances are growing. It is possible to have a positive return, averaged across the market, even in a stationary economy. The reason why is that there are really only two things a firm can do with its net profits. One possibility is that it could invest it in growing the business. However, there is not much point in doing that in a stationary economy because by assumption no increase in aggregate consumption (and therefore, in the long run, aggregate production) are possible. Therefore, firms are left with only the second option, which is to pay them out to investors as dividends. Those dividends provide a return that is independent of economic growth.

Would the stock market still be a good investment in such an economy? Yes. Well, sort of. The rate of return from firms' dividend payouts will depend on investors' demand (in aggregate) for returns on their investments. Stock prices will rise or fall, causing returns to respectively fall or rise, to find that level. If your personal desire for returns is lower than the average across the investing public, then the stock market would look like a good investment. If your desired return is higher than the average, then it will look like a poor investment. The marginal investor will, of course be indifferent. The practical upshot of this is that the people who invest in the stock market in this scenario will be precisely the ones for whom the stock market is a good investment, given their personal propensity to save and desire for returns, and so forth.

Finally, you mentioned that in your scenario the GDP stagnation is due to declining population. I am less certain what this means for investment, but my first thought is that you would have a large retired population selling its investments to fund late-life consumption, and you would have a comparatively small (relative to history) working population buying those assets. This would lead to low asset prices, and therefore high rates of return. However, that's assuming that retirees need to sell assets to fund their retirement consumption. If the absolute returns on retirees' assets are large enough to fund their retirement consumption then you would wind up with relatively few sellers, resulting in high prices and therefore relatively low rates of return. It's not obvious to me which effect would dominate, and so it's hard to say whether or not the resulting returns would look attractive to the working-age population.

Theoretically, the growth in stock prices is highly dependent on GDP. Stock prices are largely determined by their earnings and earnings are largely dependent on the economy. If the economy is good, people buy more of a product which increases earnings and thus decreases the P/E ratio which makes the stock more attractive to investors which causes people to buy and the stock price to go up.

Practically, speaking there are a lot of other variables though like costs, competition, inflation etc.

All things equal though you would expect cap gains to be virtually 0 in a 0 gdp economy however however there still can be growth within industries in a 0 gdp economy. Look at the growth of the cell phone industry during the great recession as an example.

Lastly, even if there were no growth you still have dividends which frequently pay more than bonds. The market would become perfectly efficient since anyone with a decent understanding of Excel could perfectly value the entire market in under 15 minutes.

Now, hypothetically, assume that the U.S. produces no GDP growth at all in the future due to population declining. Would this mean that the stock market (index) would also stay close to the same level in the long term, making it a bad investment?

You are forgetting several key things.

Firstly, many companies in US stock market are actually internationally diversified global companies. Lots of their business comes from emerging markets, although the company is listed in US stock exchange. So, the global GDP growth would need to stop for the stocks to stop growing. US GDP growth stopping is not enough.

Secondly, stocks pay dividends. Currently, where I live, dividends are slightly above 4% generally, whereas bond yields are negative. That alone makes stocks good investments.

Thirdly, stocks are protected from inflation. If the inflation is 2%, stocks grow 2% even without any GDP growth at all!

Now, in general you are right. Stock yield is inflation + real GDP growth + dividend yield. If inflation is 2%, real GDP growth is 2% and dividend yield is 4%, stocks yield 8%. Now, if real GDP growth would suddenly stop, what would happen? If there is no change in investors' yield demand, stocks HAVE to yield 8% in the future too. The inflation surely won't increase due to changed conditions, and real GDP growth would be 0% due to the definition of GDP growth stopping. So, the only way for stocks to yield is 6% dividend yield. However, companies have a hard time increasing dividends, so the way this happens is that stock valuations drop by 1/3 = 33.333%. If a stock paid 1 EUR dividend, it used to cost 25 EUR. After the change, it pays still 1 EUR dividend but now costs 16.67 EUR.

So, in other words:

If the global GDP growth stops forever, stocks immediately drop 33.33% and then continue yielding the same in the future. It takes bit over 5 years for stock investments (with dividends) to reach their pre-drop levels. If bond yields were positive, it would take far more for stock investments to reach bond investments, but then again, where I live, bond yields are negative.

I don't see the global GDP growth stopping forever. Do you?

The world may have finite ressources, but the world is not finite as such. There are endless methods to reuse materials, optimize processes and invent new tools and methods. When a tree dies, it doesn't go to waste. It recycles and becomes nutrition for the surroundings. It may use a bit of entropy, but that should last billions of years. The same way, we can become better at reusing materials, create robots that do our work and probably create a lot of new jobs in the meantime and perhaps even make conventional jobs obsolete. But this won't reduce reduce economic growth. On the contrary.