Food for thought:
I was watching From Russia With Love, and noted how back in the day, the Turkish government was absolutely dying to show its loyalty to NATO, and perhaps, if it were lucky, even be considered a fellow European country. Kemalist ideology reigned in the government; the government was strictly secular, and Westernization policies were put in place to emphasize Turkey's European roots.
Doesn't seem like that anymore.
For about a decade or more, Turkey has increasingly pivoted away from the West diplomatically, and shirked its role in NATO. Turkey has not participated in combat operations in Afghanistan, and has not acted with its European allies against ISIS.
Turkish internal politics largely shape the Eastward shift. Ever since the early 2000s, the Kemalist President Erdogan recently claimed that Muslims discovered America before Columbus, American soldiers were attacked in Istanbul by nationalist protestors. While these attacks do not represent the Erdogan government, they nevertheless reflect the anti-Western sentiment growing in the country.
Granted, increasing Turkish animosity towards the West is partially caused by the West rejecting Turkey. Western nations criticized the Turkish government's response to the Gezi protests. But I think the Europeans are the most to blame. Turkey's ascension to the EU is also on hold, as it has been for a long time. Angela Merkel has long opposed an EU-member Turkey, preferring a "privileged partnership." In 2007, Nicolas Sarkozy said that not just anyone can join the EU, and Turkey just isn't west enough. With its bids to join the EU spurned, Turkey feels it cannot connect with the West, and must look elsewhere.
I think the two biggest issues concerning European rejection of Turkey's ascension are a) the fact that Turkey will not call what happened to the Armenians a genocide, and b) a lot of Europeans are against the idea of a 95% Muslim nation joining the Christian-majority European Union. I think had Europeans swallowed their pride/fears and made Turkey an EU member, we would not be seeing this Eastward shift.
But would Turkey quit NATO, or be kicked out?
In the short run, no. Especially with ISIS on its border, Turkey needs a safety net, and is likely willing to put aside its hurt feelings and rely on the West for protection. Still, its improved relations with Iran and the Kurds may suggest the beginning of a shift towards less reliance on NATO for protection. Turkey's government has also been calling for the ouster of al-Assad (why, I will never understand) and only NATO has the power (though not the will) to do so.
But in the long run, now that Turkish sentiment is pivoting eastward, and with Turkey's seeming lesser role in NATO, and European spurning of Turkey, could we see Turkey leaving NATO in say, 10-15 years? Who knows. I think it's a possibility.