Bitcoin's price will never be stabilized. There will always be things such as news posts, scandals, influencers which will affect Bitcoin's price. I'm pretty sure it would be absolutely impossible to stabilize Bitcoin's price.
If the acceptability increases, then at some point the prices have to stabilize. Also, the proportion of short-term investors and day traders need to decline and that of long-term investors need to increase. Gold is extremely stable, because 95% or more of the market cap is held as long-term investment. The same figure for Bitcoin would be less than 50%. That means that the proportion available for short-term trading is tens of times higher than the case with gold. And the volatility largely results from this fact.