Bitcoin will never be as stable as fiat currency, because fiat is being intentionally stabilized by central banks. They participate on foreign exchange markets and dump their own currency if it goes up too fast, or buy it if it falls too fast. This helps them to generally reduce volatility to fractions of percent per day. Obviously it's impossible with Bitcoin and at best we will see volatility levels similar to gold.
There may be a way to somehow minimize the volatility of bitcoin but we are still long way to go as it will not happen in the next few decades or up until bitcoin haven't reach its supply limit. Still we don't know what will happen as it is an unknown water for now but I guess we will not figure that out because it is estimated that bitcoin will reach its limit in the year 2140, which I doubt that we are still alive to witness that phenomenon.