No, that's not correct, in general.

When people buy into an index fund/ETF, they are buying existing shares of that fund (which are already backed by the component stocks of that index) from other people who already have them. If there are 1M shares of an index fund that tracks the S&P500 floating around out there, and you go into your brokerage account and buy 1,000 shares, you have not increased that 1M figure by 1,000. There are still 1M shares; you have just bought 1,000 shares from an existing owner (perhaps another individual investor with an eTrade account just like you) who wanted to sell them.

In a case where an index fund does have to buy more shares of the underlying components (for rebalancing purposes or whatever), they are buying shares from other people on the open market: institutional investors, hedge funds, prop traders, etc. They are not buying from the company behind the stock ticker.

Yes, companies do sometimes issue new shares to the open market in order to raise cash. But that's not a daily activity; some companies may go years (or even forever) without doing another public offering beyond their IPO. Other companies do it somewhat regularly, perhaps a couple or few times a year. And some just do it when their stock price is high and they think offering more shares would be a good deal for them.