Not exactly. There's benefit to staying private, but it's more about the time-value of money than the need to grow. Private companies can generally optimize for longer timeframes than public companies. Except for Amazon, whose shareholders seem cool with the idea of plowing all free cash flow into growth. Shareholders are often much more conservative than private owners and won't allow drastic changes. Private owners are more comfortable with variation.

Growth is necessary for all companies, public or private. If you're not growing, you're dying. That's not always true, but it's true enough -- not just in economics, but for all organisms, literal and metaphorical. It's less about absolute growth than relative growth. If your competitor is growing faster, eventually they will crowd you out of the market.