Well, technically but that would severely complicate the situation.
France is virtually an ally of Russia in the war. The UK and Germany are aiding the French, and the Russians. Even the US is providing some targeting information, at least to the French.
So, what do these (Christian) NATO nations do if (islamic) Turkey gets into a war with (Christian) Russia who the mentioned NATO nations are already allied with?
Also, who shoots down a warplane that "violates" your airspace for a few seconds? Why?
Because Syria shot down a Turkish plane that was in the airspace for a few seconds back in 2012. The words uttered by Turkey back then could be applied to Russia today.
“who shoots down a warplane that “violates” your airspace for a few seconds? Why?”
Russian intrusion has happened earlier and repeatedly and appears to be on purpose.
“Syria has long disputed Turkey’s sovereignty over Hatay province, which borders Syria’s Latakia governorate, although the international community is in agreement that the region belongs to Turkey. In 2012 a Syrian fighter jet shot down a Turkish F-4 Phantom killing the pilot in the process over the region as well.”
http://dailysignal.com/2015/11/24/why-turkey-shot-down-a-russian-fighter-jet/