It could replace fiat if the price is stable. However.. only if it replace fiat.. it will become stable. I foreseen many physical shops with 'Accept Bitcoin' in the next 20 years or so.
Bitcoin being stable or not, it will never replace fiat currencies. And there is no need for that. Bitcoin is capable enough to succeed without replacing fiat currency.
Stability will come, but it will take a serious amount of years. I really think if Bitcoin is gaining interest from new wealthy investors it will cause another period of heavy volatility.