If by stable price the OP means for the next month or two then yes. Obviously not for the long term. The price will only stabilize when the market has been saturated (user adoption is close to its full rate), that could easily take another decade of growth to happen.
The price is stabilizing right now because the bear market is over and we are at the in-between point between bear and bull markets. This seems to have started at the beginning of this month after the last lower-high of the bear market at ~7400. Likely Bitcoin will trade within this range between that last lower-high and the bottom around 6000. This might last a couple months or it might as long as to the beginning of next year. Only time will tell. My guess would be that November will signal the breakout and the slow start of the next bull market and we'll see a rise above 7500 to break the pattern of lower highs that has happened all year by the end of November and will see $8xxx Bitcoin by end of year. So it's stable in the short to medium term (2 - 6 months) because we have entered the period between the bear and the next bull market.
I wouldn't say that the bear market is over. There's a lot of traps going on right now, so its likely the price could really stay at the $6K-$7K for all we know. I do hope that November will be different as we are all tired of seeing this bullish trend. I understand that 2017 is different, but at some point we need to have a break out run and hopefully November can be the start and see the price to at least touch 5 digit mark at the end of the year.
It is certainly possible that Bitcoin takes a step lower and breaks down through the ~6000-level support it's held all year, thus continuing and deepening the bear market, but that seems extremely unlikely at this point. Bear market means the price is moving down, the price is no longer moving down as the lower-highs have ended at ~7400 meaning their is nowhere for the highs to drop to since the support is ~6000, so we are now stuck in that range until the market is done shifting and the next bull market starts.
No the price won't simply stay stable at $6-$7k. Just like in 2015 it didn't permanently stay at $200-$300. The Bitcoin market is still very small and it is still very early in the game. A long term stable price of Bitcoin will likely be closer to $600k-$700k than $6k-$7k. There is zero doubt that we are in a short term shift from bear market to bull market (if indeed the bear market is now over, which seems very likely) and within a few months $6xxx prices will be long forgotten.
If you look at the last shift from bear to bull, it took the first 10 months of 2015 to make this shift. Right now we seem to be only 3 weeks into this period. Now I don't think we need to have any fear of this stable period lasting anywhere near 10 months. The last bear market was much longer than this one (13 months compared to 9 months) and there was nothing significant to pull Bitcoin off the bottom and start the next bull market back then. Bitcoin was much less well known, was not in the news constantly as it is now. Now we have Wall St firms preparing Bitcoin services, including major consumer company backed services like how Microsoft and Starbucks are backing the NYSE parent company's exchange that is supposed to hit in November. We have ETF decisions looming every so often that will eventually be passed, we have regulations being worked on by governments, we have Coinbase going after institutional investors, and overall there is just far more interest and attention on Bitcoin than in 2015 the last time Bitcoin entered this period. So yeah I think it is likely that we'll see a break above the last lower high of ~7400 and see the next bull market starting by end of the year, or at latest early next year.